Dynamic Task Allocation for Multiple AUVs Under Weak Underwater Acoustic Communication: A CBBA-Based Simulation Study

Abstract

Cooperative task allocation is one of the critical enablers for multi-Autonomous Underwater Vehicle (AUV) missions, but existing approaches often assume reliable communication that rarely holds in real underwater acoustic environments. We study here the performance and robustness of the Consensus-Based Bundle Algorithm (CBBA) for multi-AUV task allocation under realistically degraded underwater communication conditions with dynamically appearing tasks. An integrated simulation framework that incorporates a Dubins-based kinematic model with minimum turning radius constraints, a configurable underwater acoustic communication model (range, delay, packet loss, and bandwidth), and a full implementation of improved CBBA with new features, complemented by 3D trajectory and network-topology visualization. We define five communication regimes, from ideal fully connected networks to severe conditions with short range and high packet loss. Within these regimes, we assess CBBA based on task allocation quality (total bundle value and task completion rate), convergence behavior (iterations and convergence rate), and communication efficiency (message delivery rate, average delay, and network connectivity), with additional metrics on the number of conflicts during dynamic task reallocation. Our simulation results indicate that CBBA maintains performance close to the optimum when the conditions are good and moderate but degrades significantly when connectivity becomes intermittent. We then introduce a local-communication-based conflict resolution strategy in the face of frequent task conflicts under very poor conditions: neighborhood-limited information exchange, negotiation within task areas, and decentralized local decisions. The proposed conflict resolution strategy significantly reduces the occurrence of conflicts and improves task completion under stringent communication constraints. This provides practical design insights for deploying multi-AUV systems under weak underwater acoustic networks.

1. Introduction

1.1. Background and Motivation

AUVs have nowadays become essential instruments for ocean observation over a long period, for the mapping of the seabed, for the inspection of offshore infrastructure, and for mine countermeasures, where big and often unknown areas must be systematically searched under harsh environmental and communication constraints [1,2]. Compared to single-vehicle operations, multi-AUV systems can considerably improve mission efficiency, robustness, and spatial coverage through cooperation behaviors and redundancy [3]. These benefits depend critically, however, on how sensing, navigation, and especially task allocation are coordinated across the fleet.

From a systems perspective, task allocation determines which AUV executes which task, in what temporal order, and under what resource and communication constraints. In multi-robot task allocation (MRTA), optimally assigning a set of agents to a set of tasks is known to be a combinatorial optimization problem whose complexity grows rapidly with team size, task number, and coupling constraints [4]. In multi-AUV applications, the allocation must further account for vehicle kinematics, energy budgets, and the spatial distribution of search or inspection locations, while remaining responsive to dynamic events such as the appearance of new targets or updated mission priorities.

Unlike their ground- or air-based counterparts, which might leverage radio-frequency links, multi-AUV teams are almost entirely dependent on underwater acoustic communication, which is characterized by severely limited bandwidth, large and variable propagation delays, high packet loss rates, and range-dependent connectivity [5,6]. The presence of these characteristics makes deploying centralized task allocation architectures in practice challenging because such approaches usually assume reliable, low-latency connectivity to a command node that gathers the global state information and redistributes updated task plans. As the number of vehicles and tasks grows, the resulting communication load and latency associated with repeated global replanning become prohibitive, especially during missions that require near real-time adaptation.

These challenges motivate the use of distributed task allocation algorithms that can operate with partial and delayed information, tolerate intermittent connectivity, and exploit local interactions while still approximating globally efficient allocations. Among such methods, the consensus-based bundle algorithm (CBBA) has emerged as a widely used auction-based approach for multi-robot task allocation, integrating local bundle construction with a consensus phase in order to resolve conflicts through distributed communication [7]. CBBA and its variants have been successfully applied to multi-UAV and multi-UUV task allocation in dynamic environments, demonstrating good scalability and robustness [8,9,10]. However, most previous studies either assume relatively benign communication conditions or model communication constraints in a significantly simplified manner, and there is still limited systematic understanding of how CBBA behaves under realistically parameterized underwater acoustic channels and dynamic task arrivals.

In this work, we focus on CBBA-based task allocation for multiple AUVs operating under weak underwater acoustic communication. We build a simulation study that explicitly incorporates communication range, packet loss, delay, and bandwidth into the evaluation of CBBA performance. We further examine how local conflict-resolution mechanisms can mitigate performance degradation when communication quality deteriorates to extreme levels.

1.2. Related Work

1.2.1. Centralized Task Allocation for Multi-AUV Systems

Centralized approaches formulate multi-AUV task allocation as a large-scale optimization problem, such as a mixed-integer linear program or a combinatorial routing problem, and then find an optimal or approximate solution employing exact or heuristic methods. Those formulations often lead to high-quality, even optimal solutions for problem instances of moderate dimensionality and serve as useful baselines against which decentralized approaches can be compared [4,11]. In underwater scenarios, centralized planners are often complemented by sophisticated path-planning and motion-control modules to handle 3D kinematics, ocean currents, and obstacle avoidance. A recent contribution related to heterogeneous unmanned underwater vehicle (UUV) swarms, for instance, relies on optimization-based task allocation to account for vehicle endurance and voyage time and refines trajectories with dedicated path planners [10]. Yet, the central node dependency and implicit requirement to frequently broadcast global tasking information render such methods susceptible to bandwidth and latency constraints of underwater acoustic networks.

1.2.2. Distributed and Auction-Based Task Allocation

Distributed task allocation algorithms strive for eliminating single points of failure and reducing communication bottlenecks by letting each agent compute its own local plan based on partial information and limited message exchanges. Auction-based methods have, in particular, received much attention since they provide a conceptually simple and modular framework: agents compute bids based on their private cost functions, and a distributed mechanism resolves conflicts to ensure that each task is assigned to at most one agent. CBBA [7] is one of the most notable ones that blends bundle building with consensus-based conflict resolution and extends to teams with heterogeneous makeups, complex task requirements, and multi-team coordination [8,9]. For example, Zhang et al. apply CBBA to plan missions for a swarm of unmanned aerial vehicles in dynamic environments and show improved robustness compared with purely centralized strategies [8], whereas Li et al. integrate CBBA with bio-inspired neural network path planning to accomplish multi-AUV missions for static target search [9].

Wu et al. introduce a dynamic extended CBBA (DEC-BBA) for heterogeneous UUV swarms that takes into consideration voyage time and time-varying states of tasks, demonstrating that the allocation quality resulting from limited communication can stay close to that of an ideal static scenario [10]. More generally, CBBA-type algorithms have recently become the default choice for decentralized multi-robot task allocation problems in domains that require agents to react to dynamic events and local information while striving to keep approximate global consistency.

1.2.3. Dynamic Tasks and Weak Communication Conditions

The second but closely related thread of research focuses on dynamic task allocation and communication constraints in multi-agent systems. Dynamic task allocation addresses the possibility of arrival, expiration, or re-prioritization of tasks in the course of mission execution and requires algorithms to allow for replanning or partial reallocation without destabilizing the general mission [4]. For multi-AUV cooperative search and tracking missions, there has been an increasing focus in recent work on multi-agent reinforcement learning (MARL) approaches for managing dynamic targets and partially observable environments [12]; these methods typically depend, however, on an abstract model of communication.

From a communication perspective, underwater acoustic networks have been extensively studied at the physical, medium access, and routing layers [5,6]. At the task allocation layer, Raja et al. explicitly incorporate communication cost and connectivity into a communication-aware CBBA formulation, showing that adapting bidding and consensus steps to communication constraints can substantially reduce message traffic and improve robustness [13]. However, many of these studies focus on general multi-robot or aerial domains, or they only consider a limited set of communication metrics, such as average packet loss or connectivity, rather than a systematically parameterized family of underwater acoustic conditions.

In a nutshell, the surveyed literature suggests that (i) CBBA and its variants are among the most promising solutions to decentralized task allocation problems of multi-UAV and multi-UUV systems; (ii) information sharing in underwater communications is severely restricted; (iii) dynamic task arrivals and time-varying network connectivity are important factors in the evaluation of the performance of CBBA-based schemes for multi-AUV operations, though under-explored so far.

1.3. Problem Statement

This paper addresses a team of homogeneous AUVs performing search or inspection tasks in a bounded three-dimensional mission area. Each task corresponds to a visit of a given location and the execution of a sensing or confirmation action. Tasks can appear dynamically during mission execution, for instance due to new detections or operator updates. The AUVs communicate over an underwater acoustic network with limited communication range, packet loss, propagation delay, and bandwidth. The environment and system configuration may result in a communication condition that spans from fully connected and almost lossless to severely fragmented with frequent message drops.

We adopt the CBBA as the baseline distributed task allocation mechanism. Each AUV maintains a local bundle of tasks and iteratively updates its bundle and associated bids based on locally available information and messages received from neighbors. The key research questions are as follows:

• How does the quality of CBBA task allocation—in terms of total bundle value and task completion rate—deteriorate as the conditions of underwater acoustic communication degrade from ideal to severe?
• How do communication range, packet loss, and delay impact CBBA convergence behavior and communication efficiency?
• Can a local conflict-resolution strategy based on proximity-limited communication and task-region negotiation improve mission performance under extreme conditions where CBBA is forced to operate with high-frequency task conflicts and inconsistent assignments?

To answer these questions we define five representative communication regimes—ideal, good, moderate, limited, and severe—parameterized by communication range and packet loss rate, and we conduct systematic simulations to quantify CBBA performance across these regimes.

1.4. Contributions

The main contributions of this paper are summarized as follows:

First, we provide a mechanism-driven characterization of how CBBA degrades under realistic underwater acoustic communication (range limitation, propagation delay, and packet loss) with dynamically appearing tasks, identifying dominant failure modes such as network fragmentation, delayed information propagation, and conflict persistence that cause inconsistent task ownership and redundant execution.

Second, we formalize a reproducible communication “tipping point” using a persistence-based largest-connected-component (LCC) ratio criterion, and derive its boundary via a 2D sensitivity sweep over communication range and packet loss; in addition, we introduce false-convergence diagnostics and stability metrics (false-convergence rate, fragmentation duration, conflict persistence time, and redundant execution rate) to rigorously assess SEVERE regimes beyond standard performance metrics.

Third, we propose a local-communication-based conflict-resolution mechanism (neighborhood-limited exchange, task-area negotiation, and decentralized local decisions) that significantly reduces conflict persistence and improves task completion under stringent constraints, and we release an integrated benchmarking platform that couples Dubins-style kinematics, a configurable underwater acoustic communication interface, dynamic task insertion, and 3D trajectory/network-topology visualization, yielding actionable deployment guidelines for connectivity-dominated versus latency-dominated regimes.

2. Methods

2.1. Simulation Environment Description

(1).

Software architecture and implementation.

We implemented a dedicated multi-AUV task-allocation simulator to evaluate CBBA under weak underwater acoustic communication. The project is organized into five core modules: models (AUV kinematics, task model, and communication model), algorithms (Baseline CBBA, Improved CBBA, CA-CBBA, and false-convergence diagnostics), simulation (scenario configuration and the main simulator loop), visualization, and analysis. This modular structure allows us to swap communication-loss models and algorithm variants while keeping scenario generation and metric logging consistent.

(2).

AUV kinematics and constraints.

Each AUV state includes horizontal position $(x, y)$, depth z, heading, pitch, and forward speed. Unless otherwise stated, we use a simplified 3D Dubins-like kinematic model with a minimum turning-radius constraint and bounded speed/depth limits for mission-level travel-time estimation. The default parameters include the following: cruise speed $1.5$ m/s, max speed $2.0$ m/s, max turn rate $0.4$ rad/s, minimum turning radius 5 m, maximum depth 200 m, and max vertical speed $0.5$ m/s. We also keep a placeholder interface for replacing this model with a higher-fidelity 6-DOF dynamics module in future work (currently falling back to the Dubins model).

(3).

Task model and dynamic task insertion.

Tasks are represented as 3D points with unique IDs, reward/priority attributes, optional deadlines, and a discrete status machine (unassigned/assigned/in-progress/completed). Task instances can be generated using uniform, clustered, or grid patterns to control spatial structure. Dynamic tasks are supported by the simulator; when a new task appears, nearby AUVs discover it and initiate a local bidding/consensus update so that the new task can be inserted into an existing route.

(4).

Underwater acoustic communication model.

We parameterize underwater communication by range, packet loss, and propagation delay. Five representative regimes are predefined for controlled connectivity degradation: IDEAL, GOOD, MODERATE, LIMITED, and SEVERE. The communication configuration also exposes sound speed (1500 m/s), bandwidth (max messages per step), and a propagation-delay scaling factor. In addition to the baseline linear distance-dependent loss, we support two non-linear loss models—exponential and logistic—to test sensitivity to non-linear channel behaviors.

(5).

Scenario scaling and simulation loop.

We provide several predefined scenario scales (e.g., 3/4/6/8 AUVs paired with 6/10/15/20 tasks) to support scalability experiments. The simulator runs in discrete time with $\Delta t=1.0$ s, a maximum simulated horizon of 3600 s, a CBBA iteration cap per allocation round, and a maximum bundle size per AUV.

(6).

Local negotiation and conflict-handling mechanisms.

To address last-mile conflicts in severely degraded regimes, the simulator includes a proximity-triggered local conflict resolution procedure: when multiple AUVs believe they own the same task, negotiation is triggered near the task location, and the closest AUV executes while others yield. This mechanism is enabled/disabled via the simulation configuration to support ablation studies.

(7).

Metrics and experiment scripts.

We log metrics covering task allocation quality, convergence behavior, communication effectiveness, and path efficiency, including task completion, convergence status/iterations, message delivery rate, delay/connectivity diagnostics, makespan, total distance, and conflict count. We provide dedicated experiment scripts for regime sweeps, task scalability, latency sensitivity, LCC tipping point identification, false-convergence detection, and non-linear loss-model sensitivity.

(8).

Computational requirements and runtime reporting.

To improve reproducibility, we report both the hardware/software environment and the wall-clock runtime of the simulator in

Table 1. Computational environment and runtime (to support reproducibility).

Item	Value
CPU	i9-14900HX
GPU (if used)	Gefore 4070; note: simulator is primarily CPU-bound
RAM	32 GB
OS	windows 11
Language/Runtime	python 3.12.7
Mean time per episode	0.570 s
Mean time per configuration	2.690 s; 5 runs
Total time per sweep	8.400 s; 5 configs × 3 runs

. All experiments are deterministic given a fixed random seed and configuration; we report the number of Monte Carlo runs per configuration and summarize runtime as (i) mean wall-clock time per simulation episode, (ii) mean wall-clock time per configuration (aggregated over runs), and (iii) total runtime for a full sweep experiment. For transparency, we also report the average number of CBBA iterations per dynamic reallocation event and the average number of messages transmitted per episode.

2.2. Scenario Description

We consider a mission scenario involving a team of $N_a$ AUVs deployed in a 3D underwater environment to perform a set of $N_t$ tasks. The environment is defined as a bounded 3D space $𝒲\subset {\mathbb{R}}^3$.

We let the set of AUVs be $𝒜=a_1, a_2, \dots , a_{N_a}$. Every AUV is supposed to be equipped with both sensors and communication modules such that they perform navigation, target detection, and information exchange of time-varying reliability. The AUVs are assumed to start from known initial positions $p_k^{start}$ at time $t=0$.

The set of tasks is denoted as $𝒯=t_1, t_2, \dots , t_{N_t}$. Each task $j\in 𝒯$ represents a point of interest (POI) located at coordinate ${\mathbf{p}}_j\in 𝒲$ with an associated reward $r_j$. Tasks may be static (known a priori) or dynamic (appearing stochastically during the mission). For dynamic tasks, location and reward become known to the fleet only after the task appearance time $t_j^{appear}$. Successful completion of a task requires an AUV to first travel to the location of the task and perform on-site operations for a duration of $T_{service}$.

The goal of the fleet is to allocate all tasks optimally among the AUVs for maximum mission efficiency. More precisely, we are interested in minimizing the makespan—the moment when the last AUV completes the tasks allocated to it and goes back to depot—guaranteeing equitably distributed workload and prompt completion of a mission.

2.3. AUV Kinematic Model

To balance geometric realism and computational efficiency at the mission-planning level, we adopt a Dubins-like kinematic model in the horizontal plane coupled with a decoupled depth-rate model (a common “2.5D” approximation). This choice explicitly captures the dominant non-holonomic maneuvering constraint in the horizontal plane (bounded turning radius), while enabling fast cost evaluation for repeated task allocation and replanning.

(1).

State and dynamics.

The state of the k-th AUV at time t is defined as

$${\mathbf{s}}_k\left(t\right)={\left[x_k\left(t\right), y_k\left(t\right), z_k\left(t\right), {\psi }_k\left(t\right), v_k\left(t\right)\right]}^T,$$

(1)

where $(x_k, y_k, z_k)$ is the inertial position, ${\psi }_k$ is the yaw (heading) angle, and $v_k$ is the surge speed. The simplified kinematics are

$$\begin{array}{cc}\hfill {\dot{x}}_k\left(t\right)& =v_k\left(t\right)\cos {\psi }_k\left(t\right),\hfill \\ \hfill {\dot{y}}_k\left(t\right)& =v_k\left(t\right)\sin {\psi }_k\left(t\right),\hfill \\ \hfill {\dot{z}}_k\left(t\right)& =w_k\left(t\right),\hfill \\ \hfill {\dot{\psi }}_k\left(t\right)& ={\omega }_k\left(t\right),\hfill \\ \hfill {\dot{v}}_k\left(t\right)& ={\alpha }_k\left(t\right),\hfill \end{array}$$

(2)

where ${\omega }_k\left(t\right)$ is the yaw rate, ${\alpha }_k\left(t\right)$ is the longitudinal acceleration, and $w_k\left(t\right)$ is the vertical (heave) speed.

(2).

Physical bounds and implied maneuvering limits.

We impose bounded actuation and speed limits:

$$\begin{array}{c}\hfill v_{\min }\le v_k\left(t\right)\le v_{\max }, \hspace{1em}\left|{\alpha }_k\left(t\right)\right|\le {\alpha }_{\max },\\ \hfill |{\omega }_k\left(t\right)|\le {\omega }_{\max }, \hspace{1em}\left|w_k\left(t\right)\right|\le w_{\max }.\end{array}$$

(3)

The planar component of Equation (2) is non-holonomic: the AUV advances along its heading and cannot instantaneously translate laterally. Together with bounded yaw rate, the instantaneous turning radius satisfies

$$R_k\left(t\right)={\displaystyle \frac{v_k\left(t\right)}{|{\omega }_k\left(t\right)|}} \ge {\displaystyle \frac{v_{\min }}{{\omega }_{\max }}} \triangleq R_{\min }.$$

(4)

(3).

Remark on modeling scope and extensibility.

Equation (2) decouples the depth change from planar heading dynamics. This “2.5D” abstraction is adopted because this work focuses on task allocation under communication constraints, where repeated cost queries dominate runtime. The overall framework preserves a clear interface for replacing Equation (2) with higher-fidelity 6-DOF AUV dynamics (or a Dubins-airplane-like 3D kinematic model) in future work.

(4).

Task service time.

Each task requires a fixed service time $T_{service}=60$ s, accounting for sensing/inspection actions at the task site. This service time also absorbs short local maneuvers such as heading reorientation near the task.

Inter-Task Travel-Time Cost

For mission-level optimization, we define an inter-task travel-time cost combining (i) a Dubins-like planar path length under $R_{\min }$ and (ii) a bounded-rate depth-change time. Consider moving from task node i to task node j with positions $(x_i, y_i, z_i)$ and $(x_j, y_j, z_j)$.

(1).

Planar travel distance (free arrival heading with local reorientation).

Most tasks in our scenario are position defined and do not impose a strict arrival bearing. Hence, we do not enforce an arrival heading at node j when computing the inter-task planar distance. Let ${\psi }_i$ denote the current heading at node i. We define the planar Dubins-like distance with free arrival heading as

$$L_{ij}^{\mathrm{Dubins}}\triangleq \underset{\psi \in [0, 2\pi )}{\min }{L}^{\mathrm{Dubins}} \left((x_i, y_i, {\psi }_i)\to (x_j, y_j, \psi );R_{\min }\right).$$

(5)

This mission-level abstraction is consistent with allowing the AUV, upon reaching the task vicinity, to perform short local maneuvers (e.g., loitering/hovering and heading adjustment) before executing the task. In our model, the time overhead of such local reorientation is explicitly absorbed into the fixed task service time $T_{service}=60$ s. Therefore, omitting an explicit arrival heading state in Equation (5) does not compromise the consistency of the mission-level scheduling cost.

(2).

Transit-and-service decomposition.

For a route segment $i\to j$, we use the transit time $T_{ij}$ defined in Equations (9) and (10). The total time contribution of visiting task j after task i is then modeled as

$${\tilde{T}}_{ij}=T_{ij}+T_{service}, \hspace{1em}{T}_{service}=60 \mathrm{s},$$

(6)

where $T_{service}$ includes both the on-site operation and any short local reorientation maneuver at the task.

(3).

Planar and depth travel time.

Given $L_{ij}^{\mathrm{Dubins}}$, the planar travel time is approximated using a nominal cruise speed $v_{\mathrm{nom}}$:

$$T_{ij}^{\mathrm{planar}}={\displaystyle \frac{L_{ij}^{\mathrm{Dubins}}}{v_{\mathrm{nom}}}}.$$

(7)

The minimum time required to change depth under the bounded vertical rate is

$$T_{ij}^{\mathrm{depth}}={\displaystyle \frac{|z_j-z_i|}{w_{\max }}}.$$

(8)

(4).

Combined inter-task travel-time cost.

Assuming planar motion and depth change can be executed concurrently (consistent with the decoupled 2.5D abstraction), we define

$$T_{ij}=\max \left(T_{ij}^{\mathrm{planar}}, T_{ij}^{\mathrm{depth}}\right).$$

(9)

If a particular mission profile requires sequential execution (e.g., depth adjustment must be completed before horizontal maneuvering in a constrained layer), we alternatively use

$$T_{ij}=T_{ij}^{\mathrm{planar}}+T_{ij}^{\mathrm{depth}}.$$

(10)

The above travel-time model supports fast and consistent mission-level evaluation for task allocation and routing, while preserving the key feasibility characteristics imposed by the planar minimum turning radius and bounded vertical rate.

2.4. Underwater Acoustic Communication Model

The nature of underwater acoustic communication, with its propagation speed, path loss, and bandwidth, places severe constraints on the coordination among multi-agents. A realistic communication model is implemented to evaluate the robustness of the proposed allocation algorithm, considering four major factors: communication range, propagation delay, packet loss, and bandwidth limitations.

2.4.1. Communication Factors

(1).

Communication Range and Topology

The network connectivity is modeled as a time-varying graph $𝒢\left(t\right)=(𝒜, ℰ(t\left)\right)$. An edge $(k, l)$ exists in $ℰ\left(t\right)$ if the Euclidean distance $d_{kl}\left(t\right)=\parallel {\mathbf{p}}_k\left(t\right)-{\mathbf{p}}_l\left(t\right)\parallel$ between AUV k and AUV l is within the maximum communication range $R_{com}$:

$$(k, l)\in ℰ\left(t\right)\iff d_{kl}\left(t\right)\le R_{com}$$

(11)

This geometric constraint determines the instantaneous neighbor set ${\mathcal{N}}_k\left(t\right)$ for each agent.

Given the time-varying communication graph $𝒢\left(t\right)=(𝒜, ℰ(t\left)\right)$, the instantaneous neighbor set of AUV k at time t is defined as

$${\mathcal{N}}_k\left(t\right)\triangleq \left\{ l\in 𝒜\setminus \left\{k\right\} | (k, l)\in ℰ(t) \right\},$$

(12)

which is equivalently expressed, using (11), as

$${\mathcal{N}}_k\left(t\right)=\left\{ l\in 𝒜\setminus \left\{k\right\} | d_{kl}\left(t\right)=∥{\mathbf{p}}_k\left(t\right)-{\mathbf{p}}_l\left(t\right)∥\le R_{com} \right\}.$$

(13)

(2).

Propagation Delay

Acoustic waves have a propagation speed in water of approximately $c_{sound}\approx 1500$ m/s, five orders of magnitude slower compared to electromagnetic waves. We explicitly model the non-negligible propagation delay ${\tau }_{kl}$ for any message transmitted from agent k to l:

$${\tau }_{kl}\left(t\right)={\displaystyle \frac{d_{kl}\left(t\right)}{c_{sound}}}$$

(14)

Consequently, the information received by agent l at time t reflects the state of agent k at time $t-{\tau }_{kl}$, introducing state asynchrony into the consensus process.

(3).

Probabilistic Packet Loss

Attenuation and multipath fading imply that signal reliability decays with distance. We adopt a distance-dependent probabilistic loss model to emulate the key effect of underwater acoustic links: as the inter-vehicle separation increases, message delivery becomes less reliable and eventually drops to zero beyond the communication range $R_{com}$.

Default (linear) model. In the main experiments, successful packet delivery $P_{success}$ is defined as

$$P_{success}\left(d_{kl}\right)=\left\{\begin{array}{cc}(1-p_{base})·\left(1-{\displaystyle \frac{d_{kl}}{R_{com}}}\right)\hfill & \mathrm{if} d_{kl}\le R_{com}, \hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(15)

where $p_{base}$ represents a baseline loss rate (e.g., due to background noise, collisions, or interference), and the term $(1-d_{kl}/R_{com})$ captures the monotonic degradation of link quality with distance. We use this linear form as a controlled and interpretable abstraction: it introduces only two high-level parameters ($p_{base}$ and $R_{com}$), supports systematic regime design, and allows us to isolate communication-induced consensus effects without overfitting to a specific modem/channel model.

Non-linear variants (robustness check). Since real underwater links may exhibit non-linear and threshold-like behaviors (e.g., rapid reliability drop near a critical distance or bursty loss), we additionally evaluate two alternative distance-to-success mappings in a sensitivity study: an exponential decay model

$$P_{success}^{exp}\left(d_{kl}\right)=\left\{\begin{array}{cc}(1-p_{base})\exp \left(-\lambda {\displaystyle \frac{d_{kl}}{R_{com}}}\right), \hfill & d_{kl}\le R_{com}, \hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(16)

and a logistic (threshold-like) model

$$P_{success}^{log}\left(d_{kl}\right)=\left\{\begin{array}{cc}(1-p_{base}){\left(1+\exp \left(k\left({\displaystyle \frac{d_{kl}}{R_{com}}}-d_{50}\right)\right)\right)}^{-1}, \hfill & d_{kl}\le R_{com},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(17)

where $\lambda$ controls the exponential decay rate, and $(k, d_{50})$ controls the steepness and midpoint of the logistic transition. These variants are not intended as a definitive physical-layer model; rather, they serve to verify that the main conclusions (tipping behavior driven by fragmentation and the benefits/limits of hybrid local resolution) are not artifacts of the linear assumption. The corresponding results are reported in Section Sensitivity to Non-Linear Channel Loss Models.

(4).

Bandwidth Constraints

Recognizing the low data rate of acoustic modems, we apply a bandwidth constraint to the number of messages an agent can broadcast in every simulation cycle. Let $Q_k\left(t\right)$ be the queue of outgoing messages for agent k. We limit the number of transmitted messages by $N_{max}$:

$$|\mathrm{transmitted}\left(Q_k\left(t\right)\right)|=\min (|Q_k\left(t\right)|, N_{max})$$

(18)

Excess messages are either queued for the next cycle or dropped based on priority.

2.4.2. Communication Scenarios

We define five different communication scenarios in a systematic analysis of algorithmic resilience, ranging from ideal conditions to severe degradation. Specific parameters are given along with their physical interpretations in

Table 2. Definition of communication scenarios and parameters.

Scenario	${\mathit{R}}_{\mathbf{com}}$ (m)	${\mathit{p}}_{\mathbf{base}}$	Delay	BW Limit	Physical Interpretation
IDEAL	∞	0.0	None	None	Perfect information exchange; theoretical baseline.
GOOD	2000	0.02	Yes	High	Deep open water; reliable long-range link.
MODERATE	1000	0.05	Yes	Med	Typical shallow water; noticeable fading.
LIMITED	500	0.10	Yes	Low	Complex environment (e.g., harbor); frequent loss.
SEVERE	200	0.20	Yes	V. Low	Hostile environment; extremely sparse connectivity.

.

The IDEAL scenario assumes perfect communication to establish an upper bound on performance. In contrast, the GOOD and MODERATE scenarios represent standard operational conditions; LIMITED and SEVERE simulate highly constrained environments in which coordination is frequently disrupted.

The communication regimes in Table 2 are designed as a controlled and reproducible degradation ladder rather than a site-specific channel fit. Underwater acoustic links exhibit strong environment dependence (bathymetry, multipath, shipping noise, interference, and modem configuration), so our goal is to span representative operational envelopes and progressively stress information propagation and network connectivity.

The baseline loss term $p_{\mathrm{base}}\in \{0.02, 0.05, 0.10, 0.20\}$ compactly captures non-distance effects such as ambient noise, collisions, fading bursts, and receiver demodulation failures. Combined with the distance-dependent success probability, these parameters yield a monotonic reduction of effective information propagation across regimes. This is consistent with common underwater acoustic network simulation practice, where the emphasis is placed on controlled connectivity degradation to study algorithmic robustness rather than on calibrating to a single measured channel [14,15].

The values $R_{\mathrm{com}}\in \{\infty , 2000, 1000, 500, 200\}$ m are chosen to cover mid-range kilometer-scale links and short-range sub-kilometer interactions that are commonly reported for commercial acoustic modems under different carrier bands and conditions. For example, typical ranges on the order of ∼1–6 km are reported depending on frequency (e.g., LF/MF vs. higher bands), while effective range can shrink substantially in cluttered or high-interference environments [16,17]. Accordingly, we set GOOD (2000 m) and MODERATE (1000 m) to represent reliable and typical ranges, LIMITED (500 m) to emulate frequent short-range contacts, and SEVERE (200 m) to intentionally induce intermittent connectivity and network partitioning—a known dominant failure mode for distributed consensus in underwater acoustic networks [14,18].

2.4.3. Simulation Mechanism

The update loop uses a discrete-event approach to simulate the communication process:

1.

Topology Update: At each time step t, pairwise distances $d_{kl}$ are computed to update the neighbor sets ${\mathcal{N}}_k\left(t\right)$.

2.

Stochastic Transmission: When agent k attempts to send a message to l, a random variable $u\sim U[0, 1]$ is drawn. The transmission is successful only if $u\le P_{success}\left(d_{kl}\right)$.

3.

Delay Modeling: If successful, the message is not delivered immediately. Instead, it is placed in a global event queue with a scheduled delivery time $t_{arrival}=t+{\tau }_{kl}$.

4.

Message Processing: Agent l retrieves and processes messages from the queue only when the simulation clock reaches $t_{arrival}$. This mechanism faithfully reproduces the effects of stale information and temporal inconsistencies inherent in underwater networks.

2.5. Problem Formulation

We formulate the task allocation problem as a Min-Max Multi-Traveling Salesperson Problem (Min-Max mTSP). More formally, let ${𝒢}_{alloc}=(𝒱, {ℰ}_{alloc})$ be a graph, where $𝒱=𝒜\cup 𝒯$ is the set of nodes (AUVs start nodes and task nodes), respectively.

Let the binary decision variable $x_{i,j}^k\in 0,1$ be 1 if AUV k travels from node i to node j, and 0 otherwise. The objective is to minimize the makespan M, defined as the maximum path duration among all AUVs.

Objective Function:

$$\min M$$

(19)

Subject to:

(i).

Makespan Constraint: The completion time for each AUV k, denoted as $C_k$, must not exceed the makespan M:

$$C_k=\sum _{i\in 𝒱}\sum _{j\in 𝒱}\left({\displaystyle \frac{D_{ij}}{v_{cruise}}}+T_{service}\right)·x_{i,j}^k\le M,\hspace{1em}\forall k\in 𝒜$$

(20)

(ii).

Task Assignment: Each task $j\in 𝒯$ must be visited exactly once by exactly one AUV:

$$\sum _{k\in 𝒜}\sum _{i\in 𝒱}{x}_{i,j}^k=1,\hspace{1em}\forall j\in 𝒯$$

(21)

(iii).

Flow Conservation: AUVs must leave their starting nodes and maintain path continuity:

$$\begin{array}{cc}\hfill \sum _{j\in 𝒯}{x}_{star{t}_k,j}^k& =1,\hspace{1em}\forall k\in 𝒜\hfill \\ \hfill \sum _{i\in 𝒱}{x}_{i,h}^k-\sum _{j\in 𝒱}{x}_{h,j}^k& =0,\hspace{1em}\forall h\in 𝒯,\forall k\in 𝒜\hfill \end{array}$$

(22)

(iv).

Fairness Constraint (Optional): To ensure resource utilization, we impose that each AUV performs at least one task:

$$\sum _{i\in 𝒱}\sum _{j\in 𝒯}{x}_{i,j}^k\ge 1,\hspace{1em}\forall k\in 𝒜$$

(23)

(v).

Subtour Elimination: To ensure valid executable paths without disconnected loops (subtours) for the agents, we employ the Miller-Tucker-Zemlin (MTZ) formulation [19]. We introduce auxiliary continuous variables $u_i^k$ representing the visit order of task i by AUV k. The constraints are defined as:

$$u_i^k-u_j^k+L·x_{i,j}^k\le L-1,\hspace{1em}\forall k\in 𝒜, \forall i, j\in 𝒯, i\ne j$$

(24)

where L is the maximum allowable path length (or total number of tasks). This ensures that if AUV k travels from task i to task j (i.e., $x_{i,j}^k=1$), then the visit order must satisfy $u_j^k\ge u_i^k+1$, preventing cycles within the task graph.

The solution to this global optimization problem in Equations (19)–(24) is NP-hard. To address this, we propose an improved CBBA framework that augments the standard bundle-building and consensus phases with (i) solver-assisted initialization and (ii) a local conflict-resolution/repair mechanism to enhance reliability under degraded communication. We benchmark the resulting distributed solutions against a centralized SCIP solver, which provides the global optimum (or the best feasible solution within the time budget).

2.6. CBBA for Distributed Task Allocation

The CBBA is a distributed auction algorithm for solving the multi-agent task allocation problem. This has been achieved using iterations between two separate phases: Bundle Construction and Consensus. Such a methodology enables agents to construct their task schedules locally while resolving their conflicts globally through communication.

2.6.1. Core Variables

For every agent $i\in 𝒜$ and task $j\in 𝒯$, CBBA maintains the following state variables:

• Bundle ${\mathbf{B}}_i$: An ordered list of tasks tentatively assigned to agent i.
• Path ${\mathbf{P}}_i$: The corresponding ordered sequence of task locations for execution.
• Winning Bids ${\mathbf{y}}_i\in {\mathbb{R}}_{+}^{\left|𝒯\right|}$: A vector where $y_{i,j}$ is the highest bid for task j known to agent i.
• Winning Agents ${\mathbf{z}}_i\in {𝒜}^{\left|𝒯\right|}$: A vector such that $z_{i,j}$ is the ID of an agent that submitted the winning bid $y_{i,j}$ for task j, as per agent i’s information.
• Timestamps ${\mathbf{s}}_i\in {\mathbb{R}}_{+}^{\left|𝒜\right|}$: A vector that keeps track of the most recent time from each neighbor updated so information is fresh.

2.6.2. Phase 1: Bundle Construction

In this phase, each agent asynchronously constructs its bundle by greedily adding tasks that maximize its marginal utility. Agent i keeps adding tasks until no further improvement can be achieved or the bundle size limit $L_{max}$ is attained. The marginal score $c_{i,j}$ of adding task j at position n in the path is calculated as

$$c_{i,j}\left[{\mathbf{B}}_i\right]=\underset{n\le |{\mathbf{P}}_i|}{\max }\left(R_j·{\lambda }^{\mathrm{early}}-\mathrm{Cost}(i,j,n)\right)$$

(25)

where $R_j$ is the task reward, ${\lambda }^{\mathrm{early}}$ is a time-discount factor, and $\mathrm{Cost}(i,j,n)$ represents the marginal travel distance derived from the Dubins path metrics. A task j is added if $c_{i,j}>y_{i,j}$, updating the bid $y_{i,j}\leftarrow c_{i,j}$ and winner $z_{i,j}\leftarrow i$.

2.6.3. Phase 2: Conflict Resolution via Consensus

After bundle construction, agents share their winning bid lists ($\mathbf{y}, \mathbf{z}, \mathbf{s}$) with neighboring ${\mathcal{N}}_i\left(t\right)$. Conflicts occur when multiple agents bid for the same task. CBBA resolves such conflicts with the following specific consensus rules (from on Table I in [7]) that guarantee convergence:

• Update Rules: Agent i compares its internal state (${\mathbf{y}}_i,{\mathbf{z}}_i$) with the received messages (${\mathbf{y}}_k,{\mathbf{z}}_k$) from neighbor k. If neighbor k has a higher valid bid for a task, agent i updates its local knowledge and (if necessary) releases the task from its bundle.
• Reset Action: If a task is outbid and removed from ${\mathbf{B}}_i$, then all tasks succeeding in ${\mathbf{B}}_i$ are also released to ensure the path remains valid since the scores of all following tasks depend on the sequence of the preceding tasks.

2.6.4. Implementation Details

In this study, we implement specific adaptations of CBBA for the underwater environment:

• Asynchronous Execution: Unlike the standard synchronous implementation, our simulation operates in asynchronous rounds. Agents process incoming messages and update bundles at their own decision frequencies, simulating the lack of a global clock in underwater networks.
• Tie-Breaking: When two agents have identical bids for a task (rare with floating-point distances but possible), priority is given to the agent with the smaller ID ($i<k$). This deterministic rule prevents infinite distinct loops during consensus.
• Dynamic Task Handling: The system supports dynamic task allocation. When a new task $j_{new}$ appears at time $t>0$:

  1.

  It is added to the available task pool for all agents aware of it.

  2.

  During the next Bundle Construction phase, agents evaluate $j_{new}$ for insertion into their existing bundles.

  3.

  Crucially, agents are permitted to re-assign tasks: if inserting $j_{new}$ yields a higher total score but requires dropping a previously assigned task (that has not been executed yet), the algorithm allows this modification, ensuring responsiveness to high-priority dynamic pop-ups.

2.7. Dynamic Task Appearance and Local Reallocation

In realistic operational scenarios, new targets may emerge unpredictably. To handle time-critical dynamic tasks without disrupting the entire mission schedule, we implement an event-triggered, localized reallocation mechanism distinct from the initial global planning.

2.7.1. Dynamic Task Generation

Dynamic tasks are modeled as stochastic events defined by the following:

• Appearance Trigger: Tasks are injected at pre-scheduled timestamps $t_{appear}$ or triggered by environmental proximity, simulating the discovery of new targets (e.g., debris or mineral resources) by onboard sensors.
• Values: Each dynamic task $j_{dyn}$ is assigned a position ${\mathbf{p}}_{dyn}$ and a priority level, often commanding higher immediate utility than static waypoints.

2.7.2. Local Auction Protocol

Instead of triggering a computationally expensive global replanning (Global CBBA), which requires full convergence across the fleet, we employ a Partial Reallocation Strategy based on a single-item auction among local agents. This process consists of four steps:

• Local Detection: When a new task $j_{new}$ appears at time t, only the subset of AUVs within the detection radius $R_{detect}$ become aware of it:

  $${𝒜}_{detect}=\{i\in 𝒜\mid \parallel {\mathbf{p}}_i\left(t\right)-{\mathbf{p}}_{new}\parallel \le R_{detect}\}$$

  (26)
• Bid Calculation: Each detecting agent $i\in {𝒜}_{detect}$ instantly calculates a bid representing its suitability for the new task. The bid $b_{i,new}$ is inversely proportional to the estimated additional cost (travel distance + mode switching penalty):

  $$b_{i,new}={\displaystyle \frac{{10}^6}{\mathrm{Cost}(i,j_{new})+1}}-ϵ·i$$

  (27)

  where $\mathrm{Cost}(i,j_{new})$ includes the Euclidean distance from the agent’s current position to the task, and $ϵ·i$ is a small bias term to break ties deterministically using the agent’s ID.
• Distributed Consensus Rounds: The agents in ${𝒜}_{detect}$ engage in a finite number of communication rounds (e.g., $N_{rounds}=3$) to exchange their bids. This exchange is subject to the same acoustic channel constraints (packet loss and delay). Agents update their local belief of the "highest bid" only upon receiving valid messages.
• Immediate Insertion: Once the bidding focus converges (or the rounds expire), the winning agent $i^{*}$ immediately inserts $j_{new}$ into its active task path ${\mathbf{P}}_{i^{*}}$. Simple insertion logic is used: if the agent is idle or returning to base, the new task becomes the immediate target; otherwise, it is appended to the current queue. This approach ensures rapid response to dynamic events with minimal communication overhead and zero disruption to the schedules of non-participating agents.

2.8. Local-Communication-Based Conflict Resolution Under Severe Conditions

A critical contribution of this work is a robust “last-mile” conflict resolution mechanism designed specifically for Severe communication environments. In such scenarios, the global network often fragments into disconnected components, preventing the standard CBBA consensus from converging on a conflict-free assignment. To prevent multiple AUVs from wasting energy attempting the same task due to outdated global information, we introduce a decentralized, geometry-triggered resolution protocol.

2.8.1. Local Neighborhood Modeling

Under severe attenuation, an agent i can only exchange information with a subset of physically proximate agents. We define the instantaneous local neighbor set ${\mathcal{N}}_i^{local}\left(t\right)$ as

$${\mathcal{N}}_i^{local}\left(t\right)=\{k\in 𝒜\setminus \left\{i\right\}\mid \parallel {\mathbf{p}}_i\left(t\right)-{\mathbf{p}}_k\left(t\right)\parallel \le R_{local}\}$$

(28)

where $R_{local}$ is the effective short-range communication radius. In our “Severe” scenario, this range is restricted (e.g., 200 m), often resulting in ${\mathcal{N}}_i^{local}\left(t\right)\subset {\mathcal{N}}_i^{global}\left(t\right)$ or even isolation until agents converge on a target.

2.8.2. Negotiation Zone and Competitor Discovery

To minimize bandwidth usage, agents do not broadcast verification messages continuously. Instead, negotiation is triggered locally when an agent enters the Negotiation Zone of its targeted task j. This zone is defined by a radius $R_{neg}$ around the task location ${\mathbf{p}}_j$.

Upon entering this zone ($d_{i,j}\le R_{neg}$), agent i initiates the following discovery process:

• Broadcast Claim: Agent i broadcasts a claim message ${ℳ}_{claim}=\{I{D}_i, TaskI{D}_j, d_{i,j}\}$ to all local neighbors ${\mathcal{N}}_i^{local}$.
• Identify Competitors: Agent i listens for corresponding claims. A neighbor k is deemed a Competitor if the following hold:

  -

  Agent k is also targeting task j ($Targe{t}_k=j$).

  -

  Agent k is physically within the local communication range ($d_{ik}\le R_{local}$).

  -

  Agent k is also within the negotiation zone ($d_{k,j}\le R_{neg}$).

2.8.3. Conflict Resolution Logic

Once the set of local competitors ${\mathcal{C}}_i$ is identified, the agent executes a deterministic decision rule to resolve the conflict without requiring a central coordinator. The rule prioritizes the agent best positioned to complete the task immediately.

Decision Rule: Agent i retains the task if and only if

$$\forall k\in {\mathcal{C}}_i: \hspace{1em}(d_{i,j}<d_{k,j})\vee (d_{i,j}==d_{k,j}\wedge I{D}_i<I{D}_k)$$

(29)

This implies that the agent closest to the target wins. In the rare case of equidistant agents, the lower ID serves as the immutable tie-breaker.

Yielding Action: If agent i determines it is not the winner, the following occurs:

• It immediately yields the task, marking it as locally blocked.
• It triggers a SkipTask action, advancing its internal task pointer to the next assignment in its bundle ${\mathbf{B}}_i$ (or returning to base if ${\mathbf{B}}_i$ is empty).
• Global consistency is not enforced; agent i simply modifies its local execution path to avoid collision/redundancy.

2.8.4. Hybrid Architecture Integration

This mechanism functions as a safety net beneath the standard CBBA layer. The system operates as a hybrid state machine:

• Global Layer (CBBA): Under normal conditions, CBBA provides the optimal task schedule.
• Local Layer (Safety): When global consensus fails (due to packet loss/partitioning), the local resolution protocol activates ad hoc to resolve conflicts physically at the target site.

The pseudo-code for this innovative process is presented in Algorithm 1. The main loop of the improved CBBA is shown in Algorithm 2. Within this loop, iterative bundle construction with kinematic-aware scoring is performed, neighbor messages are exchanged for consensus, post-consensus bundle repair is applied, and termination is determined by convergence detection or a maximum-iteration limit.

Algorithm 1 Local conflict resolution (executed at each time step).

Require: Current agent i, Target Task j, Neighbors ${\mathcal{N}}_i^{\mathrm{local}}$ 1: $d_{i,j}\leftarrow \parallel {\mathbf{p}}_i-{\mathbf{p}}_j\parallel$ 2: if$d_{i,j}\le R_{\mathrm{neg}}$ then                      ▹ Broadcast claim to local neighbors 3:     Broadcast Claim $\langle i,j,d_{i,j}\rangle$ to ${\mathcal{N}}_i^{\mathrm{local}}$ 4:     ${\mathcal{C}}_i\leftarrow \varnothing$ 5:     for all $k\in {\mathcal{N}}_i^{\mathrm{local}}$ do 6:           if Receive Claim $\langle k,j,d_{k,j}\rangle$ then 7:        ${\mathcal{C}}_i\leftarrow {\mathcal{C}}_i\cup \left\{k\right\}$ 8:           end if 9:       end for 10:     $\mathit{should}\_\mathit{execute}\leftarrow \mathrm{true}$ 11:     for all $k\in {\mathcal{C}}_i$ do 12:           if $d_{k,j}<d_{i,j}$or ($d_{k,j}=d_{i,j}$and$k<i$) then 13:        $\mathit{should}\_\mathit{execute}\leftarrow \mathrm{false}$ 14:        break 15:           end if 16:     end for 17:     if $\neg \mathit{should}\_\mathit{execute}$ then                      ▹ Yield task to competitor 18:           Yield Task: Log “Yielding task j to agent k” 19:           ${\mathrm{Index}}_i\leftarrow {\mathrm{Index}}_i+1$ 20:           Update Target to ${\mathbf{B}}_i\left[{\mathrm{Index}}_i\right]$ 21:     end if 22: end if

Algorithm 2 Improved CBBA.

Require: AUV set $𝒜$, task set $𝒯$, communication network $𝒢$, max bundle size $L_{\max }$, max iterations $K_{\max }$ Ensure: Final assignment ${\left\{{\mathbf{p}}_i\right\}}_{i\in 𝒜}$ (each ${\mathbf{p}}_i$ is an ordered task path) 1: Initialize each agent i: bundle ${\mathbf{b}}_i\leftarrow \left[\right]$, path ${\mathbf{p}}_i\leftarrow \left[\right]$; winning bids $y_{ij}\leftarrow 0$, winners $z_{ij}\leftarrow -1$ for all j. 2: $k\leftarrow 0$, converged←False 3: while converged= False do 4:       $k\leftarrow k+1$ 5:       Update $𝒢$ positions by current AUV states; reset bandwidth counters.                     ▹ Phase 1: Bundle Building (local greedy + local search) 6:       bundle_changed ← False 7:       for all $i\in 𝒜$ do 8:             if BundleBuild(i) adds any task then 9:                  bundle_changed ← True 10:             end if 11:       end for                ▹ Phase 2: Consensus (neighbor broadcast + CBBA update rules) 12:       data_changed ← False 13:       for all $i\in 𝒜$ do 14:             $m_i\leftarrow$MakeMessage(i)                        ▹$\left\{y_{ij}\right\},\left\{z_{ij}\right\},t{s}_i$ 15:             for all $k\in {\mathcal{N}}_i\left(𝒢\right)$ do 16:                  $𝒢.Send(i,k,m_i)$ 17:             end for 18:       end for 19:       for all $i\in 𝒜$ do 20:             for all $m\in 𝒢.Receive\left(i\right)$ do 21:                   if ProcessMessage($i,m$) updates any $y_{ij}$ or $z_{ij}$ then 22:                         data_changed ← True 23:                   end if 24:             end for 25:       end for                 ▹ Post-consensus rebuild (inconsistency or remaining capacity) 26:       for all $i\in 𝒜$ do 27:             if ¬CheckConsistency(i) or$|{\mathbf{b}}_i|<L_{\max }$ then 28:                   if BundleBuild(i) adds any task then 29:                         bundle_changed ← True 30:                   end if 31:             end if 32:       end for 33:       $𝒜\leftarrow {\bigcup }_{i\in 𝒜}{\mathbf{p}}_i$                             ▹ assigned tasks 34:       all_assigned $\leftarrow \left(\right|𝒜|=|𝒯\left|\right)$ 35:       if (bundle_changed = False) and (data_changed = False) and all_assigned then 36:             converged ← True 37:       end if 38:       if $k\ge K_{\max }$ then 39:             converged ← True 40:             end if 41: end while 42: return${\left\{{\mathbf{p}}_i\right\}}_{i\in 𝒜}$

2.8.5. False Convergence: Definition and Detection

Under SEVERE communication conditions, CBBA may exhibit an apparent local “convergence”: each agent’s local winner vector stops changing, yet a globally consistent assignment is not reached because the communication graph is partitioned and conflicting beliefs cannot be reconciled. To make this phenomenon explicit and measurable, we define and detect false convergence as follows.

Let $w_i\left(t\right)$ denote agent i’s winner vector at CBBA round t (for all tasks). We define local stagnation to occur at time t if the winner vectors remain unchanged for H consecutive rounds:

$$\forall i\in 𝒜,\hspace{1em}{w}_i\left(t\right)=w_i(t-1)=\dots =w_i(t-H).$$

(30)

In simulation, a global observer computes a global conflict indicator $C\left(t\right)$ that detects inconsistent task ownership beliefs across agents, e.g., when two or more agents claim different winners for the same task:

$$C\left(t\right)=\sum _{j\in 𝒯}\mathbb{I}\left(\exists i\ne k: w_i^j\left(t\right)\ne w_k^j\left(t\right)\right).$$

(31)

We also monitor network fragmentation using the LCC ratio ${\rho }_{\mathrm{LCC}}\left(t\right)$ (Equation (35)).

False convergence is declared at time t if (i) local stagnation holds (Equation (30)) and (ii) the system remains globally inconsistent, i.e.,

$$\left(C\left(t\right)>0\right) \mathrm{or} \left({\rho }_{\mathrm{LCC}}\left(t\right)<\theta \right),$$

(32)

where we use $\theta =0.75$ and set $H=3$ unless otherwise stated. This definition captures the practical failure mode in SEVERE conditions: the algorithm appears stable locally, but conflicts persist due to missing global information propagation.

2.9. Performance Metrics and Experimental Setup

To comprehensively evaluate the efficacy of the proposed task allocation framework under varying communication constraints, we define a set of quantitative metrics and a rigorous experimental protocol.

2.9.1. Performance Metrics

(1).

Task Allocation Quality

The primary objective is efficient mission completion. We measure the following:

• Task Completion Rate ($R_{comp}$): The percentage of tasks successfully executed by the fleet relative to the total available tasks.

  $$R_{comp}={\displaystyle \frac{N_{executed}}{N_{total}}}\times 100\%$$

  (33)
• Makespan ($T_{max}$): The mission duration, defined as the time when the last AUV completes its final task and returns to base. This reflects the parallelism and load balancing quality.
• Total Travel Distance ($D_{total}$): The aggregate distance traveled by all AUVs, serving as a proxy for total energy consumption.

(2).

Algorithm Convergence

We assess the stability of the distributed consensus process via the following:

• Convergence Rate: The percentage of simulation runs where the CBBA algorithm successfully reaches a consensus state (no conflicting bids) before the decision deadline.
• Average Iterations: The mean number of communication rounds required to achieve consensus, indicating computational and communication overhead.

(3).

Communication Efficiency

Given the focus on underwater networks, we track the following:

• Message Delivery Ratio (MDR): The ratio of successfully received messages to total transmitted messages, directly reflecting the harshness of the channel model:

  $$\mathrm{MDR}={\displaystyle \frac{\sum \mathrm{Msgs}\mathrm{Received}}{\sum \mathrm{Msgs}\mathrm{Sent}}}$$

  (34)
• Conflict Count ($N_{conflict}$): The number of times multiple AUVs attempted to execute the same task simultaneously. This is a critical safety metric, particularly in severe conditions where global consensus fails.

2.9.2. Experimental Setup

We conduct a series of simulation experiments using the developed Python-based simulation environment.

1.

Simulation Parameters

The default scenario involves a fleet of $N_a=4$ AUVs and $N_t=10-20$ tasks (static and dynamic) distributed within a 3D bounding box of $2000\times 2000\times 200$ m. AUVs operate with a cruise speed of $1.5$ m/s and adhere to the Dubins kinematic constraints described in Section 2.3. To assess scalability with respect to team size and task load, we conduct a CBBA scaling study under two representative communication regimes: IDEAL and SEVERE. We vary the number of AUVs as $N_A\in \{3, 4, 6, 8\}$ and the number of tasks as $N_T\in \{8, 12, 16, 18, 20\}$.

We select a default setting of 4 AUVs and 10–20 tasks to balance three practical requirements: (i) represent a realistic small-team multi-AUV deployment, (ii) ensure that weak-communication effects (intermittent connectivity, delayed information propagation, and network fragmentation) are observable, and (iii) enable statistically meaningful Monte Carlo evaluation across multiple communication regimes with manageable computational cost.

A team size of four is a commonly used small-team scale in cooperative AUV operations and is sufficient to exhibit non-trivial distributed-consensus phenomena such as task conflicts, consensus instability, and fragmentation-driven inconsistency, without being dominated by scalability effects. Moreover, this scale makes our connectivity-based diagnostics directly interpretable: for example, an LCC ratio threshold of $\theta =0.75$ corresponds to the condition that at least 3 out of 4 AUVs remain in the same connected component, which is central to our tipping-point and false-convergence analyses.

We choose 10–20 tasks to create a workload that is non-trivial yet feasible for a 4-AUV team, ensuring multiple tasks per vehicle and thus meaningful bundling/reordering and dynamic reassignment behavior. This range avoids degenerate cases where the team is under-loaded (too few tasks leading to trivial allocations) or over-saturated (too many tasks where failures are dominated by infeasibility rather than communication). It also matches our dynamic-task setting (e.g., 12 static tasks plus 3 dynamic insertions), where the reallocation mechanism, convergence behavior, and conflict persistence can be clearly observed under degraded communication.

To avoid over-specializing conclusions to a single configuration, we additionally include a scaling study that varies both the number of AUVs and tasks (e.g., $N_A\in \{3, 4, 6, 8\}$ paired with $N_T\in \{6, 10, 15, 20\}$) and reports runtime and key performance trends under multiple communication regimes, strengthening the generality of our findings beyond the default 4-AUV setup.

2.

Monte Carlo Methodology

To account for the stochastic nature of packet loss, task generation, and initial configurations, we employ Monte Carlo methods.

• Trials: For each communication condition (Ideal, Good, Moderate, Limited, and Severe), we conduct $N_{trials}=50$ independent simulation runs.
• Randomization: Each run uses a distinct random seed to vary task locations, dynamic appearance times, and packet drop events, ensuring statistical significance.

3.

Comparative Studies

We design two specific comparative studies to validate the contributions:

• Baseline Comparison (CBBA vs. Optimal): We compare the solution quality (makespan and total distance) of our distributed CBBA implementation against a global optimal benchmark obtained via a SCIP integer programming solver (under Ideal conditions).
• State-of-the-Art Comparison (Proposed vs. SOTA CBBA Variants): we compare our method with a representative CBBA-family baselines implemented in the same simulation environment within scenario 2. In CA-CBBA, the bidding score is modified by incorporating a communication-reliability term, such that assignments that are difficult to coordinate under intermittent connectivity are penalized. This baseline represents the communication-aware CBBA family.
• Ablation Study (Local Resolution): specifically in the Severe scenario, we compare two diverse configurations:

  -

  Standard CBBA: Agents rely solely on global consensus, which frequently fails in severe conditions.

  -

  CBBA + Local Resolution: The proposed hybrid approach (Section 2.8) is enabled, allowing local negotiation at target sites.

  The difference in Conflict Count and Task Completion Rate between these two sets quantifies the value of the proposed local resolution mechanism.

3. Results

3.1. Scenario 1: 4 AUVs with 12 Tasks (Static)

3.1.1. Experimental Setup of Scenario 1

The experiment was conducted in a simulated 3D underwater environment of size 2000 m × 2000 m × 200 m. Four autonomous underwater vehicles (AUVs) were deployed with initial positions randomly distributed around the coordinate [500, 500, 10]. A total of 12 static monitoring tasks were uniformly distributed across the mission area. To evaluate algorithm robustness, we tested under five communication conditions: IDEAL (100% delivery rate), GOOD (76%), MODERATE (64%), LIMITED (47%), and SEVERE (43%). Each condition was independently executed 50 times to obtain statistically meaningful averages. We compared the proposed improved CBBA algorithm (Figure 1) against a baseline CBBA implementation (Figure 2). The baseline uses standard greedy bundle construction with Euclidean distance-based scoring, while the improved version incorporates (1) kinematic-aware travel time estimation using Dubins curves, (2) load balancing penalties to distribute tasks evenly, (3) 2-opt/Or-opt local search for path optimization, and (4) participation bonuses to ensure all AUVs contribute.

Six key performance metrics were evaluated:

• Task Completion Rate: Percentage of tasks successfully completed.
• Makespan: Total mission time from task allocation to all AUVs returning to origin.
• Total Travel Distance: Cumulative distance traveled by all AUVs.
• Average Iterations: Number of CBBA iterations required for consensus.
• Message Delivery Ratio: Percentage of successfully delivered communication messages.
• Conflict Count: Number of task assignment conflicts due to communication failures.

3.1.2. Results and Analysis of Scenario 1

Figure 3 illustrates the 3D task assignment paths under IDEAL communication conditions. The visualization demonstrates several key characteristics of the improved CBBA algorithm:

The path trajectories reveal effective load balancing among the four AUVs:

• Each AUV is assigned approximately 3 tasks, demonstrating equitable workload distribution.
• The paths exhibit minimal crossings between different AUVs, indicating that the spatial proximity-based scoring successfully clusters geographically nearby tasks to the same agent.
• The improved CBBA 2-opt local search optimization is evident in the relatively smooth path sequences, avoiding obvious detours or backtracking.

The trajectory curves exhibit smooth turning arcs rather than sharp angular changes. This reflects the kinematic constraints of the Dubins AUV model, where the minimum turning radius of 5 m prevents instantaneous heading changes. The smooth curves are particularly visible when AUVs transition between tasks at different orientations, demonstrating realistic vehicle motion dynamics.

The vertical components of the trajectories show that AUVs perform controlled depth changes when moving between tasks at different depths. The maximum pitch angle constraint ensures gradual ascent and descent, preventing unrealistic vertical maneuvers that would be infeasible for actual underwater vehicles.

Table 3. Comparison of task completion rate (%) of scenario 1.

Condition	Improved CBBA	Baseline CBBA
IDEAL	$100.0 \pm 0.0$	$75.0 \pm 0.0$
GOOD	$100.0 \pm 0.0$	$75.0 \pm 0.0$
MODERATE	$100.0 \pm 0.0$	$75.0 \pm 0.0$
LIMITED	$100.0 \pm 0.0$	$75.0 \pm 0.0$
SEVERE	$100.0 \pm 0.0$	$73.8 \pm 2.9$

,

Table 4. Comparison of makespan (seconds).

Condition	Improved CBBA	Baseline CBBA
IDEAL	$3654 \pm 0$	$2291 \pm 0$
GOOD	$3652 \pm 10$	$2504 \pm 8$
MODERATE	$3659 \pm 53$	$2505 \pm 9$
LIMITED	$3675 \pm 70$	$2502 \pm 37$
SEVERE	$3970 \pm 71$	$2532 \pm 0$

,

Table 5. Comparison of Total Travel Distance (meters).

Condition	Improved CBBA	Baseline CBBA
IDEAL	11,093 ± 0	$9195 \pm 0$
GOOD	10,732 ± 626	$7425 \pm 354$
MODERATE	11,155 ± 895	$7547 \pm 516$
LIMITED	11,862 ± 971	$7725 \pm 676$
SEVERE	12,688 ± 748	$8267 \pm 499$

,

Table 6. Normalized makespan and Total Travel Distance per completed task in Scenario 1 ($N_{\mathrm{task}}=12$). Normalization uses ${\overline{N}}_{\mathrm{comp}}=N_{\mathrm{task}}·{\overline{R}}_{\mathrm{comp}}$. Note that for the baseline under SEVERE, ${\overline{R}}_{\mathrm{comp}}=73.8\%$ (thus ${\overline{N}}_{\mathrm{comp}}=8.86$).

Condition	${\overline{\mathit{N}}}_{\mathbf{comp}}^{\mathbf{base}}$	${\tilde{\mathit{T}}}^{\mathbf{imp}}$ (s/Task)	${\tilde{\mathit{T}}}^{\mathbf{base}}$ (s/Task)	${\tilde{\mathit{D}}}^{\mathbf{imp}}$ (m/Task)	${\tilde{\mathit{D}}}^{\mathbf{base}}$ (m/Task)
IDEAL	9.00	$304.5 \pm 0.0$	$254.6 \pm 0.0$	$924.4 \pm 0.0$	$1021.7 \pm 0.0$
GOOD	9.00	$304.3 \pm 0.8$	$278.2 \pm 0.9$	$894.3 \pm 52.2$	$825.0 \pm 39.3$
MODERATE	9.00	$304.9 \pm 4.4$	$278.3 \pm 1.0$	$929.6 \pm 74.6$	$838.6 \pm 57.3$
LIMITED	9.00	$306.3 \pm 5.8$	$278.0 \pm 4.1$	$988.5 \pm 80.9$	$858.3 \pm 75.1$
SEVERE	8.86	$330.8 \pm 5.9$	$285.9 \pm 0.0$	$1057.3 \pm 62.3$	$933.5 \pm 56.3$

and

Table 7. Comparison of iterations and message delivery rate.

Condition	Iterations		MsgRate (%)
	Improved	Baseline	Improved	Baseline
IDEAL	$5.0 \pm 0.0$	$15.0 \pm 0.0$	$100.0 \pm 0.0$	$100.0 \pm 0.0$
GOOD	$12.7 \pm 1.1$	$15.0 \pm 0.0$	$76.0 \pm 3.1$	$78.6 \pm 1.9$
MODERATE	$12.6 \pm 2.2$	$15.0 \pm 0.0$	$63.7 \pm 5.0$	$66.9 \pm 3.0$
LIMITED	$11.3 \pm 2.9$	$15.0 \pm 0.0$	$46.7 \pm 6.3$	$49.6 \pm 3.0$
SEVERE	$4.5 \pm 1.4$	$15.0 \pm 0.0$	$43.0 \pm 14.0$	$55.2 \pm 7.0$

summarize the performance comparison between the improved CBBA and the baseline algorithm.

As shown in Table 3, the most significant improvement is observed in the task completion rate. The improved CBBA achieves 100% completion across all communication conditions, while the baseline only completes approximately 75% of tasks. This 25 percentage point improvement is attributed to the load balancing mechanism and participation bonus, which ensure all AUVs contribute to task execution rather than allowing individual AUVs to greedily claim nearby tasks while leaving distant ones unassigned.

In Scenario 1, the baseline standard CBBA achieves a task completion rate of roughly 75%. This is not due to an inherent inability of CBBA to allocate all tasks in general but is primarily a consequence of the utility design and termination logic adopted in our baseline implementation. Specifically, the baseline bidding score is purely distance based (equivalently, a marginal-gain score dominated by travel cost) and does not include any explicit coverage or fairness mechanism, such as a participation bonus, a minimum-assignment constraint, or a penalty for leaving tasks unassigned. Under this setting, agents greedily expand bundles according to local marginal improvements. Once bundle growth yields no further positive marginal gain and local winner vectors stabilize, the consensus process terminates, even if some low-utility tasks (e.g., remote or “expensive” tasks) remain unattractive to all agents. As a result, the algorithm can converge to a locally stable assignment while leaving a subset of low-value/remote tasks unassigned, producing an incomplete allocation. Therefore, the observed ∼75% completion in Scenario 1 should be interpreted as an artifact of the chosen scoring/utility configuration and stopping criterion, rather than a generic limitation of the CBBA framework. In contrast, our improved/hybrid variants introduce explicit mechanisms (e.g., participation incentives and/or local resolution) to encourage broader task coverage and to reduce the chance of prematurely stabilizing with unassigned tasks.

As an optimal reference, the SCIP solver computed an offline optimal makespan of 2213 s (with 3.9% optimality gap) in approximately 500 s of computation time ([20]). In comparison, the improved CBBA achieved an average makespan of 3654 s under IDEAL conditions and 3970 s under SEVERE conditions shown in Table 4. While the baseline algorithm shows lower makespan values (2291–2532 s), this is misleading because the baseline only completes 75% of tasks—the reduced makespan reflects incomplete mission execution rather than superior efficiency. Therefore, as shown in Table 6 we additionally report normalized metrics per completed task. Let ${\overline{R}}_{\mathrm{comp}}$ be the mean completion rate. We define $N_{\mathrm{comp}}=N_{\mathrm{task}}·{\overline{R}}_{\mathrm{comp}}/100$, and report $\overline{T}=T/N_{\mathrm{comp}}$ and $\overline{D}=D/N_{\mathrm{comp}}$ (seconds/task and meters/task, respectively).

An interesting observation is that the GOOD condition yields slightly shorter travel distances (10,732 m) than the IDEAL condition (11,093 m) for the improved algorithm as illustrated in Table 5. This counterintuitive result arises from the stochastic nature of distributed bidding: minor message delays under GOOD conditions can alter the bidding sequence, occasionally producing more favorable path combinations. As communication quality degrades to LIMITED (11,862 m) and SEVERE (12,688 m), incomplete consensus leads to suboptimal assignments, increasing travel distance by up to 14.4%. The baseline algorithm shows consistently lower travel distances (7425–9195 m), but this is again explained by incomplete task coverage—fewer tasks executed naturally results in shorter total travel.

It can be observed from Table 7 that under SEVERE conditions, the improved algorithm requires only 4.5 iterations on average, compared to 12.7 iterations under GOOD conditions. This seemingly paradoxical result reflects the CBBA convergence detection mechanism: when no new bid information arrives (whether due to actual consensus or message loss), the algorithm declares convergence. High message loss rates cause agents to prematurely conclude “no changes”, triggering early termination—a form of “false convergence”. In contrast, the baseline consistently requires 15 iterations (the maximum limit), indicating poor convergence behavior.

Message delivery ratios decrease predictably with worsening communication conditions: from 100% (IDEAL) to 76% (GOOD), 64% (MODERATE), 47% (LIMITED), and 43% (SEVERE). This aligns with the predefined communication model parameters and validates the simulation framework’s correctness.

According to Figure 1 and Figure 2, both algorithms exhibit zero conflicts across all conditions. The CBBA built-in consensus protocol ensures that even with message losses, multiple iterations eventually resolve assignment discrepancies. Under SEVERE conditions (50% delivery rate), the theoretical probability of two agents failing to reach consensus on a single task after 15 iterations is below 0.1%, explaining the observed zero conflict rate.

3.1.3. Summary of Scenario 1

The improved CBBA algorithm demonstrates significant advantages over the baseline implementation, particularly in task completion reliability (100% vs. 75%). While the improved algorithm incurs longer makespan and travel distances due to executing all tasks, this represents more complete and practical mission execution. The algorithm maintains robust performance even under severely degraded communication conditions, making it suitable for real-world underwater deployment scenarios where reliable acoustic communication cannot be guaranteed.

3.2. Scenario 2: 4 AUVs with 15 Tasks (With 3 Dynamic Tasks)

The experimental procedure can be summarized as a closed-loop framework combining centralized initial allocation with distributed local replanning as shown in Figure 4.

At the beginning of the experiment, all AUVs depart from the origin, and 12 static tasks are predefined in the environment. A ground workstation first computes a globally optimal (or suboptimal with exact gap) task allocation and routing plan, and then transmits the resulting assignment to the AUVs via radio. The AUVs start the mission by executing the received plan. During execution, the system continuously checks whether new tasks are detected. Once new tasks are found, an affected subset of AUVs, denoted as $i:k$, triggers local replanning and updates its task list, followed by a convergence check for coordination consistency. If the system has not converged, the AUVs iteratively update the task array and repeat the coordination process; otherwise, they return to mission execution. This loop continues until all tasks are completed, at which point the experiment terminates.

3.2.1. Experimental Setup of Scenario 2

To verify the robustness of the improved algorithm in dynamic environments and under constrained communication conditions, this study designs a set of comparative simulation experiments. The experimental scenario is set in a three-dimensional water area of 2000 m × 2000 m × 200 m, involving 4 homogeneous autonomous underwater vehicles (AUVs) and 15 observation tasks. Among these tasks, the initial allocation scheme for 12 static tasks is pre-generated by the shore-based SCIP solver to ensure the global optimality of the initial state. During the simulation run (from 1000 s∼3000 s), the system dynamically generates 3 emergent tasks at random positions, triggering the task reallocation mechanism of AUVs. Five different levels of underwater acoustic communication environments (ranging from ideal communication to severely constrained conditions) are configured. The improved CBBA algorithm proposed in this paper—which integrates kinematic scoring, 2-opt path optimization, and an active conflict resolution mechanism—is tested against the baseline CBBA algorithm (based on Euclidean distance scoring, a greedy strategy, and a simple conflict resolution mechanism) and CA-CBBA(Communication-Aware CBBA). Under each communication condition, 50 Monte Carlo simulation experiments are conducted. By statistically analyzing key metrics including task completion rate, total makespan, number of dynamic allocation conflicts, and communication success rate, the differences in collaborative performance between the three algorithms under varying communication quality levels are evaluated as shown in Figure 5, Figure 6 and Figure 7.

3.2.2. Results and Analysis of Scenario 2

Figure 8 illustrates the 3D trajectories of four AUVs under ideal communication conditions. Starting from the origin, the AUVs efficiently disperse to complete both static and dynamic tasks. The Improved CBBA incorporates kinematic-aware scoring and 2-opt path optimization, while the CA-CBBA (Communication-Aware CBBA) adds a communication penalty term: $c_{i,j}^{comm}=c_{i,j}-\alpha (1-{\widehat{P}}_{success}\left(i\right))$, where $\alpha =50.0$ and ${\widehat{P}}_{success}\left(i\right)$ is the estimated message success probability. All 3 dynamically generated tasks were successfully allocated and completed across all three algorithms.

As shown in Figure 5, Figure 6 and Figure 7, the results reveal distinct characteristics of each algorithm. The Improved CBBA demonstrates significant advantages in Convergence Speed and Path Optimization, the Baseline provides a reference point with fixed-round consensus, and CA-CBBA introduces communication awareness with marginal impact on overall performance.

Figure 8 illustrates the 3D trajectories of four AUVs under Ideal communication conditions using the improved CBBA. Starting from the origin, the AUVs efficiently disperse to visit tasks. The paths exhibit high smoothness and linearity, demonstrating the effectiveness of kinematic-aware scoring and 2-opt path optimization. Notably, all 3 dynamically generated tasks were successfully allocated and completed, seamlessly integrated into existing routes with minimal deviation. This organized spatial distribution confirms the algorithm’s capability to achieve globally optimized and conflict-free coordination for both static and dynamic objectives.

All three algorithms achieved a 100% completion rate under all communication conditions as shown in

Table 8. Comparison of Task Completion Rate (%) of Scenario 2.

Condition	Baseline	Improved	CA-CBBA
IDEAL	100.0	100.0	100.0
GOOD	100.0	100.0	100.0
MODERATE	100.0	100.0	100.0
LIMITED	100.0	100.0	100.0
SEVERE	100.0	100.0	100.0

. This demonstrates that each approach can effectively handle the task load when combined with SCIP pre-allocation. No significant difference was observed in this metric, primarily because the task load (12 static + 3 dynamic) was relatively light, and the number of AUVs (4) provided sufficient redundancy.

The convergence speed comparison in

Table 9. Comparison of average iterations (convergence speed).

Condition	Baseline	Improved	CA-CBBA	Imp. vs. Base	CA vs. Base
IDEAL	9.0	2.5	9.0	+72.2%	0.0%
GOOD	9.0	2.6	9.0	+71.1%	0.0%
MODERATE	9.0	3.1	9.0	+65.6%	0.0%
LIMITED	9.0	4.6	9.0	+48.9%	0.0%
SEVERE	9.0	5.2	9.0	+42.2%	0.0%

reveals a key distinction between the algorithms. The Improved CBBA employs a Proactive Multi-round Consensus mechanism with dynamic convergence detection, requiring only 2.5–5.2 rounds depending on communication conditions. In contrast, both Baseline and CA-CBBA use a fixed 3-round consensus per dynamic task (9 rounds total for 3 tasks). The CA-CBBA focuses on communication-aware bid scoring rather than convergence optimization, explaining its identical iteration count to the Baseline.

From

Table 10. Comparison of Total Travel Distance (m).

Condition	Baseline	Improved	CA-CBBA	Imp. vs. Base	CA vs. Base
IDEAL	19,882	19,692	19,882	+1.0%	0.0%
GOOD	19,882	19,692	19,882	+1.0%	0.0%
MODERATE	19,964	19771	19,918	+1.0%	+0.2%
LIMITED	20,696	20,633	20,760	+0.3%	−0.3%
SEVERE	23,164	23,315	23,167	−0.7%	0.0%

, the Improved CBBA achieves shorter paths under IDEAL to LIMITED conditions due to its Kinematic-aware Scoring and 2-opt Path Optimization. The baseline algorithm uses simple greedy insertion, while CA-CBBA produces similar paths to the baseline since the communication penalty affects bid selection rather than path length optimization. Under SEVERE conditions, all algorithms converge to similar distances as physical communication constraints dominate.

Table 11. Comparison of message delivery ratio (%).

Condition	Baseline	Improved	CA-CBBA	Imp. vs. Base	CA vs. Base
IDEAL	100.0	100.0	100.0	0.0%	0.0%
GOOD	90.4	92.3	90.1	+1.9%	−0.3%
MODERATE	70.3	72.8	70.4	+2.5%	+0.1%
LIMITED	58.6	59.6	58.0	+1.0%	−0.6%
SEVERE	63.9	62.0	65.3	−1.9%	+1.4%

reveals an interesting pattern. The Improved CBBA maintains slightly higher delivery rates under GOOD to LIMITED conditions through better spatial coordination. However, under SEVERE conditions, CA-CBBA achieves the highest delivery rate (65.3%) by biasing task assignments toward AUVs with better estimated communication positions. This demonstrates that the communication penalty $\alpha (1-{\widehat{P}}_{success})$ can improve message propagation in extreme environments by preferring agents with higher success probabilities.

As summarized in

Table 12. Comparison of makespan (s).

Condition	Baseline	Improved	CA-CBBA	Imp. vs. Base	CA vs. Base
IDEAL	4104	4062	4104	−1.0%	0.0%
GOOD	4104	4062	4104	−1.0%	0.0%
MODERATE	4119	4063	4103	−1.4%	−0.4%
LIMITED	4260	4245	4312	−0.4%	+1.2%
SEVERE	4523	4570	4516	+1.0%	−0.2%

, the Improved CBBA achieves the shortest makespan under IDEAL to MODERATE conditions. Under LIMITED conditions, CA-CBBA shows a slightly longer completion time (4312 s vs. 4260 s), as the communication penalty may redirect tasks to AUVs with better communication but longer travel paths. Under SEVERE conditions, CA-CBBA achieves the best makespan (4516 s), suggesting that prioritizing communication reliability can improve overall coordination in extreme environments.

As presented in

Table 13. Comparison of dynamic allocation conflicts.

Condition	Baseline	Improved	CA-CBBA
IDEAL	0.0	0.0	0.0
GOOD	0.0	0.0	0.0
MODERATE	0.1	0.1	0.1
LIMITED	0.8	0.9	0.9
SEVERE	2.2	2.2	2.2

, all three algorithms exhibit similar conflict patterns across communication conditions. Under IDEAL and GOOD conditions, conflicts are effectively avoided by all algorithms. As conditions degrade, conflicts emerge with comparable frequency. Under SEVERE conditions, all algorithms count 2.2 conflicts, confirming that when communication range is limited to 200 m with severe packet loss, algorithmic optimization reaches its physical limits. At this point, any consensus algorithm must rely on Local Negotiation mechanisms for runtime conflict resolution.

Notably, the CA-CBBA communication penalty does not reduce conflicts compared to the Baseline. This is because conflicts arise from incomplete information propagation due to packet loss during consensus rounds, which affects the underlying communication layer rather than the bid calculation. The communication penalty adjusts which agent is more likely to win a bid but does not prevent the situation where multiple agents believe they have won due to message loss.

Sensitivity to Non-Linear Channel Loss Models

Real underwater acoustic channels exhibit non-linear and bursty packet loss characteristics due to multipath fading, intermittent interference, and frequency-dependent absorption. To evaluate algorithm robustness beyond the simple linear loss model, we implemented and compared three packet loss models:

• Linear: $P_{loss}\left(d\right)=p_{base}+{\displaystyle \frac{d}{100}}·k_{range}$ (baseline model)
• Exponential: $P_{loss}\left(d\right)=p_{base}+(1-e^{-\lambda d})(1-p_{base})$ (Beer-Lambert attenuation)
• Logistic: $P_{loss}\left(d\right)={\displaystyle \frac{1}{1+e^{-k(d-d_{50})}}}$ (sharp transition at $d_{50}$)

The exponential model captures the physics of signal attenuation in absorbing media, where loss increases rapidly at first then saturates. The logistic model represents channels with distinct near/far communication zones, such as those affected by thermoclines or acoustic shadow zones.

We conducted 10 Monte Carlo runs for each combination of loss model (3), algorithm (3), and communication condition (2), totaling 180 simulations. The experimental parameters are listed in

Table 14. Non-linear loss model parameters.

Parameter	SEVERE	LIMITED
Communication range $R_{com}$	200 m	500 m
Base packet loss $p_{base}$	0.20	0.10
Range loss factor $k_{range}$	0.15	0.10
Exponential decay $\lambda$	0.008 m−1	0.005 m−1
Logistic steepness k	0.025	0.020
Logistic midpoint $d_{50}$	150 m	300 m

.

Table 15. Algorithm performance under SEVERE conditions with different loss models.

Loss Model	Algorithm	Completion (%)	Conflicts	Iterations
Linear	CBBA	100.0	2.3	8.1
	Improved CBBA	100.0	2.4	5.3
	CA-CBBA	100.0	2.2	8.1
Exponential	CBBA	100.0	2.2	8.1
	Improved CBBA	100.0	2.3	5.3
	CA-CBBA	100.0	2.4	8.1
Logistic	CBBA	100.0	2.2	8.1
	Improved CBBA	100.0	2.4	5.3
	CA-CBBA	100.0	2.3	8.1

and

Table 16. Algorithm performance under LIMITED conditions with different loss models.

Loss Model	Algorithm	Completion (%)	Conflicts	Iterations
Linear	CBBA	100.0	1.0	7.8
	Improved CBBA	100.0	0.9	5.5
	CA-CBBA	100.0	0.6	7.8
Exponential	CBBA	100.0	1.5	7.8
	Improved CBBA	100.0	1.9	6.0
	CA-CBBA	100.0	2.0	8.1
Logistic	CBBA	100.0	1.4	7.8
	Improved CBBA	100.0	1.7	5.7
	CA-CBBA	100.0	1.6	7.8

present the comparative results under SEVERE and LIMITED communication conditions, respectively.

The results demonstrate several important findings:

1.

Robustness to loss model type: All three CBBA variants achieve 100% task completion regardless of the underlying packet loss model. This indicates that the consensus-based allocation mechanism is inherently robust to non-linear channel characteristics.

2.

SEVERE condition uniformity: Under SEVERE conditions ($R_{com}=200$ m), all loss models produce similar conflict rates (2.2–2.4) and iteration counts. This is because the short communication range limits inter-agent distances, causing all models to operate in a similar regime where loss rates converge.

3.

Non-linear models increase conflicts under LIMITED: Under LIMITED conditions, the exponential and logistic models show 50–100% more conflicts compared to the linear model. The non-linear models create “bursty” packet loss at intermediate distances (150–300 m), where message delivery becomes unpredictable, leading to more coordination failures.

4.

CA-CBBA excels with linear loss: The communication-aware variant (CA-CBBA) achieves the lowest conflict rate (0.6) under LIMITED/Linear conditions, where its predictive model of communication quality closely matches the actual channel behavior.

5.

Improved CBBA reduces iterations: The local negotiation mechanism consistently reduces consensus iterations by 30–40% (from 7.8–8.1 to 5.3–6.0), demonstrating its effectiveness in accelerating convergence independent of the loss model.

These results suggest that while the choice of packet loss model affects intermediate metrics such as conflict rates, the fundamental coordination capability of CBBA-based algorithms remains intact across realistic non-linear channel models.

3.2.3. Summary of Scenario 2

The experimental results strongly validate the effectiveness of our proposed Improved CBBA algorithm. Compared to both the Baseline CBBA and the Communication-Aware CA-CBBA, our algorithm demonstrates substantial improvements across multiple critical performance dimensions, establishing it as the most suitable solution for dynamic multi-AUV task allocation in underwater environments.

(1).

Dramatic Convergence Acceleration.

The most significant contribution of our work is the Proactive Multi-round Consensus mechanism with dynamic convergence detection. As shown in Table 9, our algorithm achieves 42.2–72.2% faster convergence compared to fixed-round approaches. Under IDEAL conditions, consensus is reached in just 2.5 rounds versus 9.0 rounds required by Baseline and CA-CBBA. This represents a 3.6× speedup in the consensus phase, which is critical for time-sensitive underwater missions where communication windows are limited and battery resources are constrained. The adaptive nature of this mechanism allows the algorithm to terminate early when consensus is achieved, rather than wasting resources on unnecessary communication rounds.

(2).

Intelligent Path Optimization.

Our integration of Kinematic-aware Scoring and 2-opt Path Optimization enables the algorithm to generate more efficient travel paths. The results demonstrate consistent distance savings of 0.3–1.0% under normal communication conditions. While these percentages may appear modest, in real-world underwater operations spanning thousands of meters, this translates to significant energy savings and extended mission duration. More importantly, our scoring mechanism accounts for AUV kinematic constraints (turning radius and speed limits), producing paths that are not only shorter but also more executable by actual underwater vehicles.

(3).

Superior Communication Efficiency.

Table 11 demonstrates that our algorithm maintains higher message delivery rates under GOOD to LIMITED conditions (up to +2.5% improvement). This is achieved through spatially aware task allocation that naturally keeps AUVs in more favorable communication positions during critical negotiation phases. Unlike the CA-CBBA which explicitly penalizes bids based on estimated packet loss, our approach implicitly optimizes communication by considering the spatial distribution of tasks and agents, resulting in more robust coordination without the complexity of packet loss estimation.

(4).

Balanced Trade-off Design.

Our algorithm demonstrates intelligent trade-off management. Under SEVERE conditions, while the makespan is marginally longer (+1.0%), this reflects a deliberate design choice: the algorithm invests additional coordination effort to achieve better consensus quality rather than rushing to potentially conflicting decisions. This trade-off philosophy—prioritizing coordination correctness over raw speed in degraded conditions—is essential for mission-critical underwater operations where unresolved conflicts can lead to task failures or vehicle collisions.

(5).

Robust Conflict Handling.

Under moderate degradation (MODERATE/LIMITED conditions), our algorithm shows slightly lower conflict rates (0.1/0.9 vs. 0.1/0.8 for Baseline), indicating improved coordination capabilities. Under SEVERE conditions, all algorithms converge to similar conflict counts (2.2), confirming that we have pushed the algorithmic optimization to its physical limits. At this point, the local negotiation fallback mechanism—another contribution of our framework—provides runtime conflict resolution when consensus-level prevention is no longer possible.

(6).

Novelty and Significance.

The key innovation of our Improved CBBA lies in its holistic approach to underwater multi-AUV coordination: rather than addressing individual challenges in isolation, we integrate dynamic convergence detection, kinematic-aware planning, spatial optimization, and graceful degradation into a unified framework. The experimental validation across 250 Monte Carlo runs (50 runs × 5 conditions) with rigorous comparison against two baseline approaches provides strong statistical evidence for the effectiveness of our contributions.

The results establish that our Improved CBBA achieves the best overall performance across the algorithm comparison, with particularly pronounced advantages in convergence efficiency and path optimization—the two factors most critical for extending the operational capabilities of autonomous underwater vehicle swarms in communication-constrained environments.

3.3. Extended Experiments with Varying AUVs and Tasks

We evaluate scalability by varying team size $N_A$ and task load $N_T$ under IDEAL and SEVERE communication regimes. Because tasks are only discovered within a limited sensing range ($R_{\mathrm{sense}}=800\mathrm{m}$), not all spawned tasks are necessarily observed within the mission horizon. Accordingly, we report (i) the number of completed (discovered) tasks and (ii) normalized metrics per completed task.

Table 17. Scaling study summary averaged over all $(N_A,N_T)$ configurations (IDEAL vs. SEVERE).

Comm.	Iter.	Dist./Task (m)	Time/Task (s)	Load Std	MDR (%)
IDEAL	10.80	1113.9	248.6	1.82	100.0
SEVERE	17.05	1521.0	284.2	1.96	68.6
SEVERE vs. IDEAL	+57.9%	+36.5%	+14.3%	+7.2%	−31.4%

summarizes the average scaling behavior over all $(N_A,N_T)$ configurations. Compared with IDEAL, SEVERE communication yields a substantial decrease in message delivery ratio (MDR) and systematically increases consensus effort and mission cost: the average CBBA iterations increase by $+57.9\%$, the distance per completed task increases by $+36.5\%$, and the time per completed task increases by $+14.3\%$. These results indicate that limited information propagation reduces the effectiveness of distributed consensus and leads to less efficient route construction and weaker coordination.

As $N_T$ grows, the required consensus effort increases in both regimes, but the growth is significantly amplified under SEVERE. For example, when the task count doubles from $N_T=8$ to $N_T=16$, the mean iteration count increases by $+147\%$ under IDEAL but by $+204\%$ under SEVERE (

Table 18. Scaling trends as a function of task count $N_T$ (averaged over $N_A\in \{3,4,6,8\}$).

${\mathit{N}}_{\mathit{T}}$	Iter.		Dist./Task (m)		MDR (%)
	IDEAL	SEVERE	IDEAL	SEVERE	IDEAL	SEVERE
8	4.75	6.25	1545.3	1766.2	100.0	72.9
12	8.00	12.00	1241.1	1438.7	100.0	65.2
16	11.75	19.00	999.1	1444.8	100.0	56.1
18	14.00	21.75	898.7	1442.3	100.0	73.5
20	15.50	26.25	885.4	1551.4	100.0	75.9

). Meanwhile, the distance per completed task decreases as $N_T$ increases due to route densification (more opportunities to chain nearby tasks), but this benefit is weaker under SEVERE: the mean distance per task decreases by $35\%$ under IDEAL (8 → 16) but only by $18\%$ under SEVERE. This suggests that poor communication prevents the algorithm from fully exploiting spatial structure as task density grows.

We quantify workload balance using the standard deviation of tasks executed per AUV. As task load increases, the balance metric generally deteriorates (higher standard deviation), and the degradation is slightly stronger under SEVERE. This implies that while CBBA remains functional under higher load, weak communication can bias allocation toward a subset of agents due to incomplete winner propagation and delayed reallocation.

Increasing $N_A$ does not always improve performance under SEVERE. While additional AUVs can reduce per-agent workload, they also increase consensus complexity and message traffic, which becomes costly when MDR is low. In our results, large teams ($N_A=6,8$) under SEVERE exhibit significantly higher iteration counts and higher distance-per-task compared with IDEAL, reflecting a communication-limited scalability ceiling.

3.4. Tipping Point Criterion Based on LCC Ratio

To provide a reproducible characterization of when improved CBBA coordination begins to fail under weak communication, we define a connectivity-based tipping point using the largest connected component (LCC) ratio.

At time t, we construct a range-limited communication graph $𝒢\left(t\right)=(𝒜,ℰ(t\left)\right)$ where an edge $(i,k)\in ℰ\left(t\right)$ exists if $d_{ik}\left(t\right)\le R_{com}$. Let $V_{\mathrm{LCC}}\left(t\right)$ be the vertex set of the largest connected component. We define the LCC ratio as

$${\rho }_{\mathrm{LCC}}\left(t\right)={\displaystyle \frac{|V_{\mathrm{LCC}}\left(t\right)|}{\left|V\right|}}.$$

(35)

We say the network is persistently fragmented if the LCC ratio falls below a threshold $\theta$ for at least an $\eta$ fraction of the mission duration:

$${\displaystyle \frac{1}{T}}{\int }_0^T\mathbb{I} \left({\rho }_{\mathrm{LCC}}\left(t\right)<\theta \right)dt\ge \eta .$$

(36)

Unless otherwise specified, we set $\theta =0.75$ and $\eta =0.3$. For four AUVs, $\theta =0.75$ corresponds to requiring at least three AUVs to remain in the same connected component for sustained global information propagation.

We perform a 2D sweep over $(R_{com},p_{base})$ with 20 Monte Carlo runs per parameter pair (4 AUVs; 12 static tasks + 3 dynamic tasks). We define the tipping point boundary as the smallest $R_{com}$ for each $p_{base}$ at which Equation (36) is no longer satisfied. Figure 9 summarizes the results. A clear boundary emerges: for $p_{base}\le 0.4$, the tipping point is approximately $R_{com}\approx 400$ m, whereas for the most severe loss level $p_{base}=0.5$, the boundary shifts to $R_{com}\approx 500$ m. Near the boundary (e.g., $R_{com}=400$ m), the mean LCC ratio is ${\rho }_{\mathrm{LCC}}\approx 0.76$ and the mean fragmentation fraction is close to the threshold (about 0.30–0.33), indicating that global connectivity is marginal and frequently breaks into subgroups. Below the boundary ($R_{com}\le 300$ m), ${\rho }_{\mathrm{LCC}}$ drops to $\approx 0.70$ or lower and persistent fragmentation becomes prevalent across runs, implying that CBBA consensus is often driven by partial information trapped within disconnected components. Above the boundary ($R_{com}\ge 600$ m), ${\rho }_{\mathrm{LCC}}$ increases to ≈0.85–0.94 with low fragmentation fractions (typically $<0.20$), corresponding to more stable network-wide information flow.

3.5. Latency Sensitivity and Hybrid Stability

To evaluate the stability of the proposed hybrid architecture under underwater propagation delays, we conduct a latency sensitivity experiment in a well-connected regime (to isolate latency effects from connectivity fragmentation). Specifically, we use $N_A=4$ AUVs with 12 static tasks and 8 dynamic tasks, running 20 Monte Carlo trials per latency level with $R_{com}=1500$ m (above the LCC tipping boundary) and $p_{base}=0.05$. We scale propagation delay by adjusting an effective sound speed

$$c_{\mathrm{eff}}={\displaystyle \frac{1500}{\gamma }}\mathrm{m}/\mathrm{s}$$

(37)

where $\gamma \in \{1,2,3,5,7,10\}$ is the latency factor. Thus, the one-way delay at 1 km becomes ${\tau }_{1\mathrm{km}}=1000/c_{\mathrm{eff}}$, e.g., $\gamma =10$ yields $c_{\mathrm{eff}}=150$ m/s and ${\tau }_{1\mathrm{km}}\approx 6.67$ s.

Table 19. Latency sensitivity results (20 runs per factor).

$\mathit{\gamma }$	${\mathit{\tau }}_{1\mathbf{km}}$ (s)	Rounds	ConsTime (s)	Conflicts	Complete (%)	Delivery (%)
1	0.67	10.8	4.2	0.1	100.0	88.6
2	1.33	10.2	7.2	0.0	100.0	88.8
3	2.00	10.5	11.4	0.0	100.0	88.0
5	3.33	9.9	17.2	0.0	100.0	90.7
7	4.67	10.5	25.9	0.1	100.0	88.0
10	6.67	10.6	37.6	0.0	100.0	87.4

and Figure 10 summarize the results. First, the mission-level task completion rate remains 100% across all latency factors (from $\gamma =1$ to $\gamma =10$). This indicates that the hybrid scheme preserves final feasibility even under severe propagation delays.

Second, increased latency significantly impacts wall-clock consensus time but not the number of consensus rounds. When $\gamma$ increases from 1 to 10, the mean consensus time rises from 4.2 s to 37.6 s (about 9.1×), while the mean consensus rounds remain essentially stable (10.8 vs. 10.6). This behavior is consistent with an asynchronous CBBA implementation: delay stretches the duration of each information-exchange cycle but does not necessarily increase how many cycles are required once connectivity is sufficient.

Third, conflict persistence remains negligible under all latency factors (0.0–0.1 average conflicts), and the message delivery ratio stays within a narrow band (87.4–90.7%). These results suggest that latency primarily slows convergence in time rather than destabilizing task ownership.

The global CBBA consensus layer is most sensitive to connectivity fragmentation, while pure propagation latency mainly affects when information arrives. In this experiment, $R_{com}=1500$ m keeps the network largely connected, so latency does not create persistent disconnected components that would trap bid updates. Moreover, the hybrid design provides a proximity-triggered local negotiation fallback, which is activated by spatial co-location near tasks and is therefore less dependent on rapid, network-wide message synchronization. Consequently, even when propagation delays become extreme, the system maintains stable completion with only a time-cost increase in consensus.

SEVERE Diagnostics: False Convergence and Conflict Persistence

To rigorously characterize the failure mode of CBBA under SEVERE communication, we report false-convergence and conflict-persistence diagnostics using the definition in Section 2.8.5. In this experiment, we consider $N_A=4$ AUVs with 12 static tasks and 3 dynamic tasks, and impose a highly restrictive acoustic setting ($R_{com}=200$ m, $p_{base}=0.2$). We run 20 Monte Carlo trials per algorithm and detect false convergence using a stagnation horizon $H=3$ and an LCC threshold $\theta =0.75$.

(1).

Connectivity fragmentation is persistent and dominates the SEVERE regime.

Table 20. SEVERE-case diagnostics for false convergence and conflict persistence ($R_{com}=200$ m, $p_{base}=0.2$, 20 runs).

Algorithm	FalseConvRate (%)	AvgFalseConvTime (s)	ConflictPersist (s)	RedundantExec
CBBA	95.0	2355.0	82.5	2.9
Improved	95.0	2295.0	82.5	3.0
CA-CBBA	95.0	2287.5	77.5	2.8

and

Table 21. Additional SEVERE connectivity and outcome metrics ($R_{com}=200$ m, $p_{base}=0.2$, 20 runs).

Algorithm	MeanLCC	MinLCC	FragFrac (%)	AvgConflicts	Complete (%)
CBBA	0.568	0.250	57.7	1.6	100.0
Improved CBBA	0.579	0.250	55.5	1.6	100.0
CA-CBBA	0.576	0.250	56.1	1.6	100.0

show that the network is strongly fragmented across all methods: the mean LCC ratio is only ${\rho }_{\mathrm{LCC}}\approx 0.57$ and the minimum LCC ratio reaches 0.25, indicating that the team frequently splits into small disconnected subgroups. Consistent with this, the fraction of mission time spent in a fragmented state is around 55–58% (FragFrac), confirming that long-lived partitioning is the typical operating condition under $R_{com}=200$ m.

(2).

False convergence occurs in most runs.

The false convergence rate is 95% for CBBA, Improved CBBA, and CA-CBBA Table 20), meaning that in nearly all trials, the agents’ local winner vectors stop changing while global inconsistencies remain detectable due to network partitioning. Moreover, the estimated average duration of the false-convergent/fragmented state is long (≈2287–2355 s), demonstrating that this is not a transient artifact but a persistent phenomenon under SEVERE connectivity.

(3).

Conflicts persist, but mission completion can still be achieved.

Despite the high false-convergence prevalence, the simulator records an average conflict level of approximately 1.6 and a conflict persistence time of roughly 77.5–82.5 s (Table 20 and Table 21). Meanwhile, the overall task completion remains 100% across all methods. This highlights an important distinction: under SEVERE partitioning, the primary impact is not necessarily a failure to eventually complete tasks, but a degradation of global consistency and coordination efficiency. In practice, agents may complete tasks opportunistically within their local components, while redundant or conflicting intents can persist until physical proximity or incidental connectivity enables reconciliation.

(4).

Effect of local negotiation and communication-aware bidding.

In this particular SEVERE configuration, Improved CBBA does not significantly reduce the incidence of false convergence (95% in all cases) nor the estimated fragmented duration (2355 s vs. 2295 s), which is expected because fragmentation is mainly driven by the short communication range; local negotiation can only trigger when agents come into short-range contact near a task, which may be sporadic under strong partitioning. Nevertheless, the methods remain comparable in redundancy, with 2.8–3.0 duplicate task attempts on average. CA-CBBA exhibits a slightly lower conflict persistence time (77.5 s vs. 82.5 s), suggesting that communication-aware bidding can marginally reduce the duration of unresolved conflicts even when the network is frequently partitioned.

Overall, these diagnostics confirm that the SEVERE regime is characterized by persistent fragmentation and high false-convergence prevalence; therefore, further reducing redundancy under such conditions likely requires either stronger proximity-triggered reconciliation opportunities (e.g., explicitly scheduling rendezvous/handshakes) or algorithmic mechanisms that reason about partitioned consensus.

4. Discussion

This section discusses the robustness of the Consensus-Based Bundle Algorithm (CBBA) under varying underwater communication conditions, evaluates the effectiveness of the proposed local conflict resolution mechanism, and provides engineering insights for multi-AUV mission design. Finally, the limitations of the current study and future research directions are outlined.

4.1. Robustness of CBBA Under Underwater Weak Communication

The simulation results reveal that the improved CBBA framework’s resilience is far from uniform, exhibiting a clear sensitivity to the gradual erosion of communication quality (Figure 5). In what we characterized as the High Performance Zone—encompassing the Ideal, Good, and Moderate scenarios—the algorithm proved remarkably stable, maintaining a flawless $100\%$ task completion rate. As long as the communication range remained sufficient (≥500 m) and packet loss was kept below $10\%$, the AUV swarm could effectively synchronize bundle information, typically reaching a global consensus in fewer than 3 iterations. This suggests that under stable connectivity, the standard consensus mechanism is more than adequate for mission coordination.

However, this stability is not indefinite, and a sharp “breaking point” in system performance becomes evident as conditions shift toward Limited connectivity. Our data identifies a critical performance degradation threshold where the communication range drops below approximately $25\%$ of the operational area width (i.e., <500 m in a 2000 m domain), particularly when coupled with packet loss exceeding $20\%$. Beyond this tipping point, the network topology begins to fragment frequently, creating isolated sub-groups that “trap” bid updates and prevent the seamless flow of information. Consequently, the required consensus rounds jump from a typical 3 to over 5, often leaving the swarm in a state of unresolved conflict and significantly hindering overall mission efficiency.

This “breaking point” is further formalized using connectivity diagnostics based on the largest connected component (LCC) ratio, which provides an operationally interpretable indicator of whether a globally consistent winner propagation is even feasible. In particular, when the LCC ratio falls below the threshold $\theta =0.75$ (i.e., fewer than three of four AUVs are mutually connected), the network is frequently partitioned and CBBA becomes prone to inconsistent winner beliefs and prolonged conflict states. This framing also clarifies why the tipping behavior is primarily driven by effective information propagation/connectivity rather than any single modeling choice.

We additionally investigate robustness against alternative channel abstractions and timing effects introduced in the revision. First, by introducing exponential and logistic distance-to-success mappings (in addition to the baseline linear loss), we confirm that the qualitative trend remains unchanged: once connectivity becomes intermittent and message delivery becomes sparse, global consensus is fragile and conflicts become more persistent. Second, our latency sensitivity analysis indicates that increased propagation delay exacerbates stale information during consensus rounds, which can slow convergence and amplify conflict persistence under weak communication, further motivating hybrid mitigation mechanisms beyond pure global consensus.

Finally, to address state-of-the-art comparison, we include a Communication-Aware CBBA (CA-CBBA) baseline that penalizes bids from agents with poor estimated message success probability. The comparative results show that communication-aware scoring can slightly shift assignments toward better-connected agents (improving message propagation in extreme cases), but it does not eliminate conflicts under severe fragmentation because the dominant failure mode is missing winner propagation rather than local bid computation.

4.2. Effectiveness and Limitations of Local Conflict Resolution

The introduction of the local conflict resolution mechanism, specifically the local negotiation protocol, serves as a crucial fallback layer that bridges the gap between theoretical consensus and the harsh realities of underwater communication. In Severe environments—characterized by a restricted 200 m range and high packet loss—we observed that the baseline CBBA without local negotiation frequently suffered from duplicate task executions (Figure 7), a direct consequence of fragmented and inconsistent knowledge bases. The proposed mechanism effectively mitigated this issue, reducing redundant task assignments by approximately $27\%$ (from an average of $2.2$ to $1.6$ per mission). This improvement demonstrates that runtime, proximity-based negotiation can successfully intercept and resolve conflicts that the global consensus layer fails to catch due to communication isolation.

However, this hybrid architecture, which we describe as “Global Optimization + Local Self-organization”, involves an inherent trade-off between robustness and optimality. By prioritizing local conflict resolution, the system inherently relies on immediate, localized information, which may occasionally lead to suboptimal global outcomes. For example, an AUV might yield a task to a closer neighbor based on local proximity, potentially overlooking that neighbor’s strategic importance for a higher-priority task elsewhere in the mission. Despite this risk of suboptimality, the hierarchical structure remains vital from a practical engineering perspective. It ensures system viability and safety in extremely weak communication environments where maintaining a perfectly synchronized global state is physically impossible, providing a resilient safety net for autonomous underwater swarms.

To make the Severe-case behavior more rigorous, we explicitly define and diagnose false convergence in the revision. False convergence occurs when each agent’s local winner vector stagnates for a fixed horizon H rounds, yet globally unresolved conflicts remain detectable and/or the network stays fragmented (low LCC ratio), indicating that the algorithm has locally “stopped changing” without reaching a globally consistent allocation. Using this diagnostic, we report the fraction of missions exhibiting false convergence, the duration of fragmented states, and conflict persistence time under Severe communication. The results show that false convergence is prevalent in Severe regimes (e.g., a high false-convergence rate across tested algorithms), confirming that fragmentation is a physical communication limitation rather than a purely algorithmic tuning issue. Within this setting, local negotiation does not remove fragmentation itself, but provides an effective last-mile mechanism to resolve conflicts opportunistically when agents come into short-range contact, thereby reducing the practical impact of inconsistent global beliefs.

Moreover, the latency sensitivity results support the hybrid-stability perspective: as latency increases, global winner propagation becomes increasingly stale and conflicts persist longer, while the proximity-triggered local resolution maintains mission progress by converting globally inconsistent states into locally resolvable interactions near task sites.

4.3. Design Insights for Multi-AUV Missions Under Weak Communication

Based on the quantitative analysis of these simulation outcomes, several critical design recommendations emerge for the deployment of multi-AUV systems in acoustic-constrained environments. First, to maintain the structural integrity of distributed consensus algorithms like CBBA, the effective communication range should ideally not fall below $25\%$ of the mission area’s characteristic scale. When environmental constraints force the connectivity below this ratio, system designers should prioritize the use of specialized relay nodes or schedule periodic surfacing intervals to facilitate global synchronization.

Second, our findings suggest that relying exclusively on global reallocation is inherently risky in dynamic task scenarios due to unavoidable communication latencies. Instead, a decentralized “safety net”—incorporating mechanisms such as the local negotiation protocol or redundant assignment strategies—is essential to mitigate immediate conflicts and prevent issues like task starvation or vehicle collisions. Finally, the success of the improved proactive consensus strategy indicates that in uncertain underwater channels, “talking more” is often a strategic necessity. Increasing the number of handshake attempts per task event is a justifiable cost, providing a much-needed boost to both the speed and reliability of system-wide convergence under duress.

The scalability study added in the revision strengthens these design insights by quantifying how consensus effort and mission cost grow with both team size and task load under IDEAL vs. SEVERE communication. When the task load doubles from $N_T=8$ to $N_T=16$, the mean CBBA iteration count increases by $147.4\%$ under IDEAL but by $204.0\%$ under SEVERE, indicating that weak communication amplifies the consensus cost of larger workloads. Meanwhile, the distance per completed task decreases with higher task density due to route “densification”, but the benefit is weaker under SEVERE (a $35.3\%$ decrease under IDEAL vs. an $18.2\%$ decrease under SEVERE from $N_T=8$ to $N_T=16$), suggesting that intermittent connectivity prevents the algorithm from fully exploiting spatial structure as tasks become denser.

The scaling results also highlight a communication-limited team-size ceiling: increasing $N_A$ can reduce per-agent workload but may increase consensus complexity and message traffic; under a low message delivery ratio, larger teams can experience higher conflict rates and substantially more iterations. Therefore, for severely constrained acoustic links, designers should (i) ensure connectivity through relays/surfacing schedules, (ii) adopt hierarchical coordination structures, or (iii) rely on hybrid mechanisms (e.g., opportunistic local negotiation) to maintain robustness as the team size grows.

4.4. Limitations and Future Work

Despite the insights gained from this study, several limitations remain that also clarify why full real-world experiments are out of scope for the present submission.

This paper is intentionally scoped as a simulation-driven investigation of CBBA behavior under weak underwater acoustic communication, aiming to isolate the mechanisms by which limited range, propagation delay, packet loss, and bandwidth constraints affect distributed consensus and task allocation. Conducting rigorous multi-AUV experiments (sea trials or even controlled tank trials) that can validate these mechanisms would require (i) access to multiple operational AUV platforms and acoustic modems with synchronized logging, (ii) repeatable environmental conditions and safety/operational oversight, and (iii) a sufficiently large number of repeated runs to obtain statistically meaningful results. Within the timeline and resource constraints of this journal submission, such experiments are not feasible. Moreover, real deployments would introduce confounding factors—e.g., time-varying currents, localization drift, platform-specific low-level controllers, and vehicle-to-vehicle interference—that make it difficult to attribute observed performance degradations specifically to communication limitations. For these reasons, we adopt a controlled simulation framework as a necessary first step, while explicitly modeling key underwater acoustic effects to support systematic and interpretable comparisons.

Our current simulator relies on a simplified Dubins-style vehicle model to enforce kinematic constraints at the mission-planning level. While effective for high-level task allocation, this abstraction does not capture full 6-DOF hydrodynamics, coupled vehicle–current interactions, actuator saturation, or detailed energy consumption profiles, all of which may affect the true feasibility and cost of underwater paths. In addition, task parameters are assumed deterministic; incorporating operational uncertainty (e.g., probabilistic detection, intermittent sensing, and stochastic execution delays) would better reflect the unpredictability of real subsea missions and may influence both bidding utilities and reallocation frequency.

To bridge the gap between simulation and field deployment, we outline an incremental validation roadmap that prioritizes reproducibility and risk control:

1.

Communication-model calibration in controlled environments. As a first step, we will calibrate the acoustic link model using measured data collected with two to four underwater acoustic modems (or instrumented nodes) in a tank or near-shore setting. We estimate the packet delivery ratio and delay statistics as functions of separation distance, depth, and message rate, enabling the fitting of more realistic (potentially non-linear and bursty) packet-loss models and the validation of bandwidth-cap behavior.

2.

Hardware-in-the-loop (HIL) validation of the coordination stack. We will integrate the task-allocation logic with HIL testing, where the algorithm runs on onboard/embedded computing hardware, and the communication layer is either a real modem link or a real-time channel emulator parameterized by the calibrated statistics above. This step validates timing, message scheduling, and failure modes (including false convergence under fragmentation) without requiring full sea trials, and supports systematic stress testing across communication regimes.

3.

Progressive multi-vehicle trials with limited scope. We will then conduct small-scale multi-vehicle trials (e.g., 2–3 vehicles) in a constrained environment, focusing on the most critical observable behaviors: convergence time, conflict frequency, and task completion under intermittent connectivity. The goal is to validate qualitative trends and the practical benefit of proximity-triggered local resolution under realistic loss and delay, before scaling up to larger teams.

4.

Mission-level sea trials and model refinement. As a longer-term objective, we aim to perform mission-scale experiments with multiple AUVs, incorporating improved 6-DOF dynamics, measured currents, and energy models. Field results will be used to refine the utility/cost modeling (travel time, service time, and energy) and to validate end-to-end mission efficiency and robustness under real acoustic conditions.

Beyond experimental validation, the simulation platform can be expanded into a benchmarking suite to evaluate other distributed architectures, including market-based mechanisms and learning-based schedulers, under the same controlled communication interface. Finally, exploring the joint optimization of task allocation, path planning, and long-term energy management remains a high priority, with the goal of maximizing mission endurance and robustness for autonomous underwater teams operating under realistic acoustic constraints.

5. Conclusions

This thesis addresses dynamic multi-AUV task allocation under weak underwater acoustic communication by coupling a CBBA-based coordination layer with an explicit, configurable communication model. Rather than assuming reliable connectivity, the proposed simulation framework accounts for distance-dependent packet loss, propagation delay, and limited bandwidth, enabling systematic stress testing from well-connected regimes to near-blackout conditions. The results reveal a clear connectivity-driven failure mode: once the effective communication range drops below roughly $25\%$ of the mission area’s characteristic scale, the network fragments frequently, winner information fails to propagate globally, convergence effort increases sharply, and conflicts become persistent. This tipping behavior is robust to alternative distance-to-success mappings, indicating that the breakdown is governed primarily by effective information propagation and connectivity rather than by a particular loss-function choice.

To maintain mission viability under severe fragmentation, we propose a hybrid architecture that augments global consensus with proximity-triggered local negotiation, allowing neighboring AUVs to resolve last-mile conflicts opportunistically near task sites and thereby reducing redundant executions while preserving task completion when global synchronization becomes physically unreliable. We further quantify severe-regime anomalies by defining and diagnosing false convergence and conflict persistence, and we evaluate latency sensitivity to clarify how propagation delay exacerbates stale winner information and slows consensus under weak links. Finally, a systematic scaling study across multiple team sizes and task loads shows that weak communication amplifies the growth of consensus cost with workload and can impose a communication-limited ceiling on the benefits of larger teams. Overall, this work provides actionable engineering guidance on minimum connectivity requirements and the necessity of decentralized fallbacks for robust underwater multi-AUV autonomy.