When Does Domination Matter: A Structural and Computational Study of Spanning and Dominating Trees in Geometric Networks

Abstract

In geometric communication networks, a backbone is useful only if it is inexpensive to build and, at the same time, close enough to the demand points it must serve. This paper studies a backbone design problem in geometric communication networks that explicitly captures this trade-off between connectivity and user coverage. Two classical combinatorial optimization paradigms—the minimum spanning tree (MST), which promotes low-cost connectivity, and the dominating tree (DT), which additionally enforces that every node either belongs to the backbone or is adjacent to an active backbone node—are considered. To compare both paradigms within a common framework, this paper proposes a unified mixed-integer optimization model that balances backbone-construction and user-assignment costs. Three classes of exact formulations, namely MTZ, single-flow, and cut-set formulations, are developed. In particular, the single-flow model with valid inequalities and root-aware connectivity cuts is strengthened. For larger instances, the exact approach is complemented with a local branching matheuristic. Finally, theoretical results on computational complexity, formulation structure, and dominance relations between the MST and DT models are provided. Computational experiments show that the single-flow formulation achieves the best scalability. Furthermore, a sensitivity analysis with respect to the communication radius and the weighting parameter $\alpha$ reveals a structural transition: as the network becomes denser or the objective becomes more coverage-oriented, MST and DT solutions tend to converge. The results give a concrete way to identify when domination constraints are worth imposing and when a simpler spanning tree design already captures the relevant structure.

1. Introduction

The main goal of this study is related to the usefulness of domination constraints as they change with the geometry of a network. If the communication radius increases, the underlying graph becomes denser; consequently, more links become available and the set of feasible backbone topologies expands. In such settings, the additional domination requirement may become less significant since an MST-based backbone can remain sufficiently close to the rest of the network. An equivalent effect arises when $\alpha$ decreases, since the objective function places more emphasis on user service than on the cost of the backbone. This creates the basic trade-off studied in this paper. Connectivity pushes the model toward small and inexpensive trees, whilst coverage pushes it toward backbones that are more spread out and closer to users [1,2,3].

The minimum spanning tree (MST) is a classical combinatorial optimization problem that allows for the construction of a backbone. More precisely, it consists of obtaining network connectivity at a minimum cost, making it attractive for applications related to communication, routing, and data collection. However, notice that MST-based designs focus only on connectivity among selected nodes and do not explicitly consider the coverage of the remaining nodes. In contrast, the dominating tree problem, which is another combinatorial optimization problem, requires that every node either belongs to the backbone or is adjacent to at least one of the active backbone nodes. This is particularly relevant in wireless networks since dominating and connected structures are commonly used as virtual backbones for routing, aggregation, and topology control [1,2,3,4,5]. Although both paradigms are natural in network design, their structural relationship in geometric networks remains only partially explored.

Recent work confirms the high relevance of virtual backbone construction and domination-based models in modern communication networks. Quality virtual backbones have been studied under cooperative communication and faulty link conditions [1,2,3], whilst connected, weakly connected, and multi-fold dominating variants have motivated the search for increasingly sophisticated local search, GRASP, and hybrid metaheuristic approaches [4,5,6,7]. In parallel, exact and mixed-integer optimization methods for domination-related problems have also evolved, including recent work on domination in general graphs, clustered dominating trees, and tree-based Euclidean communication structures [8,9,10]. Nevertheless, a unified exact framework that jointly studies spanning tree and dominating tree backbones in a common geometric communication network setting is still lacking effort.

This paper addresses this gap by studying the design of MST and DT backbones within a unified optimization framework for geometric communication networks. The proposed models combine backbone-construction and user-assignment costs into a single objective, which is especially important in communication frameworks where backbone nodes are expected not only to ensure connectivity but also to provide efficient access to nearby users or traffic sources. The integration of user-assignment costs sets the proposed framework apart from both classical facility location and standard network design models. Facility location models usually optimize facility opening and user assignment without requiring the selected facilities to form a connected backbone. In contrast, spanning tree and network design models often optimize connectivity without explicitly accounting for user-to-backbone service costs. Here, both dimensions are optimized simultaneously: selected nodes must form a connected tree, and users must be assigned to active nodes. This simultaneous approach makes it possible to study the trade-off between infrastructure cost and service proximity in MST- and DT-based backbone designs. To analyze both paradigms, three classes of mixed-integer formulations for each model are proposed. A Miller–Tucker–Zemlin (MTZ) formulation, a single-commodity flow, and a cut-set formulation are developed and compared. These models are further strengthened by valid cuts and root-aware connectivity inequalities, including constraints that can be dynamically separated through lazy constraint callbacks [11]. This allows us to assess not only solution quality but also formulation strength, separation behavior, and computational scalability. Notice that classical MST formulations usually assume that all relevant terminals are already fixed, and classical DT formulations focus primarily on domination and connectivity. In contrast, the proposed approach jointly decides which nodes should be activated, how they should be connected, and how users should be assigned to the resulting backbone. This integration is important because it allows for comparing connectivity-driven and domination-driven designs under exactly the same cost structure and geometric assumptions.

This paper also considers the structural properties of the networks obtained. More precisely, the communication radius and the weighting parameter that affect the objective function in terms of connectivity and coverage are also studied. Notice that small radii domination constraints play an active role, forcing the activation of additional nodes, which leads to more expensive backbone infrastructures. By contrast, if the radius increases, the graph becomes denser, and the MST and DT solutions tend to converge into the same output solution. To further examine this effect, the joint impact of the communication radius and the weighting parameter through a three-dimensional sensitivity study is realized to separate the regions in which domination is structurally relevant from those in which dominance becomes irrelevant. Thus, the contribution is not only to redefine MST or DT structures. Instead, it is to place them in the same mixed-integer model, solve them endogenously with comparable formulations, and use the results to locate the regimes in which domination actually changes the selected backbone. The contribution of this paper can be understood at three complementary levels. First, the modeling level, where a unified mixed-integer framework is established in which MST- and DT-based backbones are compared under the same geometric graphs. For this purpose, a user-assignment structure and a weighted objective function are introduced. At a second level, three exact modeling families, namely MTZ, single-flow, and cut-set formulations, are considered. Then, the computational behavior changes across both backbone paradigms are analyzed. Finally, at the structural level, a unified framework to empirically identify the regimes in the $(\alpha ,r)$ parameter space where domination constraints are binding and the regimes where they become practically redundant is determined.

Consequently, the main contributions can be summarized as follows. A unified mixed-integer optimization approach is proposed for geometric backbone design that simultaneously determines node activation, backbone connectivity, and user assignment. This modeling step allows MST- and DT-based structures to be compared under the same objective function and the same geometric assumptions. Next, the MTZ, single-flow, and cut-set formulations are developed and compared for both MST and DT variants. This formulation-oriented step allows us to assess the computational strength, scalability, and practical limitations of each exact modeling strategy. Then, valid inequalities and root-aware connectivity cuts are derived and tested for the single-flow formulation, as it turns out to be the most effective exact model in our experiments. Lastly, a structural computational analysis of the $(\alpha ,r)$ parameter space is needed to identify when domination constraints change the optimal backbone and when MST and DT solutions coincide or become nearly identical in practice. Notice that the proposed approach is neither a direct reformulation of a Steiner tree nor a connected dominating set problem, nor a classical coverage–connectivity deployment model. Instead, it simultaneously combines endogenous node activation, tree connectivity, domination constraints, and user-assignment costs, allowing for a direct comparison between MST- and DT-based backbone designs.

For this purpose, the present work aims to not only solve either an MST- or a DT-based problem in isolation. Instead, it provides a unified optimization and computational framework for comparing both paradigms under the same geometric, assignment, and weighting assumptions. This makes it possible to identify not only which formulation is computationally stronger but also the operating regimes in which domination constraints are structurally active or redundant.

The remainder of this paper is organized as follows. Section 2 reviews the most relevant literature on backbone design, dominating structures, spanning trees, and geometric network models. Section 3 presents the proposed optimization formulations; here, the MTZ, single-flow, and cut-set models for both MST and DT are considered. Section 4 provides theoretical analysis, including structural properties, formulation relationships, and valid inequalities. In Section 5, the computational experiments, including formulation comparisons, the impact of valid inequalities, cut-set connectivity cuts, and the three-dimensional analysis of the interaction between the communication radius and the weighting parameter, are reported. Finally, Section 6 provides a brief discussion of the implications of the results from a network design perspective and concludes the article with directions for future research.

2. Related Work

From the existing literature, notice that there are two key open issues. First, there is limited exact optimization work that compares the MST and DT backbones within a unified model that explicitly accounts for user service. Second, although formulation studies are common in network design, they have rarely been used to analyze when domination meaningfully changes the solution and when it becomes less relevant as the network becomes denser. This is precisely the gap addressed in this paper: both paradigms within the same optimization framework are studied and examined not only for their computational behavior but also for the regimes in which their solutions begin to overlap.

The present work is also related to, but different from, classical Steiner tree, connected dominating set, and coverage–connectivity models. In Steiner tree problems, the set of terminals is usually given in advance, and the main decision is to find a minimum-cost connected subgraph spanning those terminals, possibly using additional Steiner nodes. In contrast, the proposed MST-based model does not assume a fixed terminal set: active nodes are selected endogenously through the interaction between backbone-construction costs and user-assignment costs. Connected dominating set models are close to the DT problem of our work, as they impose both connectivity and domination. However, they usually study domination-based backbones as standalone structures, whereas our approach compares DT solutions directly with MST-based solutions under the same geometric graph, user set, and objective function. Finally, coverage–connectivity models in wireless networks capture related spatial trade-offs, but they are often designed for deployment, clustering, or topology control decisions rather than for a formulation-oriented comparison of MST and DT backbones. Therefore, the novelty of this paper lies in the integration of these lines into a single exact optimization and structural analysis framework.

In wireless sensor and ad hoc networks, connected dominating sets and related structures have long been used as virtual backbones because they provide compact routing infrastructures while preserving neighborhood coverage. Recent work has revisited this topic from several perspectives. Quality virtual backbones under faulty links [1], cooperative communication backbones [2,3], and dominating set variants with stronger connectivity requirements [4,5,6,7] reflect the same underlying design perspective: reducing backbone size or cost while maintaining adequate coverage, robustness, or routing quality. In addition, exact and heuristic approaches for domination-type structures have continued to evolve in both pure graph settings and network-oriented applications [8,9,12,13,14].

A second related research stream concerns simultaneous coverage–connectivity optimization in wireless sensor networks. In this literature, the goals are always focused on deployment, clustering, or energy-aware topology construction instead of exact backbone tree design. These studies are highly relevant because they formalize the same spatial trade-off that motivates our work. As the communication graph becomes denser, connectivity becomes easier to maintain, and the need for additional relay or dominating nodes decreases. This approach emerges in deployment models with simultaneous coverage and connectivity requirements [15], infrastructure-oriented topologies and smart metering applications [16], and more recent studies on coverage/connectivity maximization and network design surveys [17,18]. However, this line of work usually stops short of comparing, within a unified exact optimization framework, how connectivity-driven and coverage-driven backbones behave as the communication radius and the connectivity–coverage trade-off parameter vary.

A third topic is directly connected to our methodological contribution: mixed-integer formulations and polyhedral strengthening for tree and network design problems. Although many articles on virtual backbones give more relevance to heuristics or distributed protocols, fewer studies provide exact optimization models that enable systematic comparisons across formulations. Concerning the literature related to network design, recent computational studies on integer programming formulations [19,20], vulnerability-aware network design [21,22], and robustness enhancement through cut-set or connectivity-based techniques [23,24] highlight the importance of formulation strength and valid inequalities. Moreover, the literature on communication-oriented spanning trees helps position our contribution more precisely. In particular, studies in [25,26] show that routing and communication costs can significantly increase model complexity and that exact and polyhedral approaches are relevant for handling such types of problems. These works are closely related to ours because they also combine tree structure, communication costs, and exact optimization. However, they do not explicitly analyze the interaction between spanning trees and domination-based backbones altogether in geometric graphs, nor do they investigate the conditions under which both structures may become equivalent.

From the domination side, recent studies have also focused on richer variants such as two-connected, twofold-connected, weakly connected, capacitated, and fault-tolerant dominating structures [4,5,6,7,13,14]. These contributions reinforce the importance of robustness and service-oriented backbone requirements, but they still tend to study dominating structures in isolation. In contrast, our work places MST and DT models within a unified framework that explicitly incorporates user-assignment costs, allowing a direct structural and computational comparison between connectivity-driven and coverage-driven backbone designs.

In general, the gap addressed in this paper is twofold. On the one hand, despite substantial progress in virtual backbone construction, domination models, and coverage–connectivity deployment, there is still limited exact optimization work jointly comparing MST and DT backbones under a common objective that explicitly accounts for user service. However, although formulation comparisons are common in network design, they have not been systematically used to study the structural convergence between domination-based and spanning tree-based backbones, as the communication radius and the connectivity–coverage trade-off parameter vary. This comparison reveals an important distinction with respect to previous works. Existing studies have focused either on domination-based virtual backbones, spanning tree formulations, or coverage–connectivity models. In contrast, the present paper explicitly connects these three perspectives by embedding MST and DT backbones in the same geometric user-assignment approach and by evaluating their computational and structural behavior over the same parameter space.

To better position our contribution within the literature,

Table 1. Positioning of the present study with respect to the most closely related literature.

References	DT/CDS	MST	Geometric	Exact MIP	Multi-Formulation	Structural Analysis
[1,2,3]	✓	✗	✓	✗	✗	✗
[4,5,6,7]	✓	✗	✗	✗	✗	✗
[8,9]	✓	✗	✗	✓	✗	✗
[10]	✗	✓	✗	✓	✗	✗
[12,13,14]	✓	✗	✗	✓	✗	✗
[19,20]	✗	✓	✗	✓	✓	✗
[15]	✓	✗	✓	✗	✗	✗
[21,22,23,24]	✗	✓	✗	✓	✓	✗
[16,17,18]	✓	✓	✓	✗	✗	✗
[25,26]	✗	✓	✗	✓	✓	✗
This paper	✓	✓	✓	✓	✓	✓

provides a structured comparison between recent related works and the present study. The main dimensions that are relevant to our setting include the backbone paradigm, geometric modeling, exact optimization, formulation diversity, polyhedral strengthening, and structural analysis. This table reveals not only the methodological differences but also the specific contributions of this paper in an attempt to bridge connectivity-driven and coverage-driven backbone design within a unified framework. The differences between these streams are relevant to positioning the present contribution. Works on connected dominating sets and virtual backbones mainly emphasize domination, robustness, or distributed construction mechanisms, but they often omit comparing the resulting structures with spanning tree-based backbones under a common objective. Conversely, studies on spanning tree and communication tree formulations provide strong exact optimization and polyhedral perspectives, but they generally do not impose domination requirements or analyze when such requirements are redundant in geometric graphs. Coverage–connectivity models in wireless networks capture related spatial trade-offs, but they are often formulated as deployment, clustering, or topology control problems rather than as a direct formulation-level comparison between MST and DT backbones. The present paper is positioned at the intersection of these lines: it combines MST and DT structures, user-assignment costs, multiple exact formulations, and an empirical structural analysis over the $(\alpha ,r)$ space.

Thus, the findings of this paper should be understood as an integrative and comparative contribution: MST and DT are not introduced as new graph-theoretic objects but are embedded in a common optimization framework that enables a systematic formulation-level and structural comparison in geometric communication networks with user-assignment costs. Table 1 mentions the main dimensions of the literature that are closely related to our study. Notice that many contributions have addressed several of these aspects separately, including domination-based backbones, spanning tree models, geometric settings, exact optimization, and formulation-oriented analyses. However, it draws our attention that these dimensions are not integrated within a single framework. In particular, there is still limited exact optimization work that unifies MST- and DT-based backbones simultaneously with a single objective function that explicitly incorporates user-assignment costs while also examining the role of geometric density and the connectivity–coverage trade-off. In this sense, it can be said that our work is positioned at the intersection of these research directions by combining both backbone paradigms, multiple exact formulations, and a structural analysis of the network regimes in which domination remains relevant or becomes irrelevant. This positioning motivates the unified modeling and analysis framework introduced in the next section.

3. Proposed Optimization Formulations

A geometric communication network defined on a set of candidate nodes V embedded in the Euclidean plane is considered. Each node represents a potential backbone location, and a set of users K must be served by the selected backbone infrastructure. The users in K represent external demand points, traffic sources, or clients that must be served by the selected backbone. Notice that they are not necessarily candidate backbone nodes. This difference separates infrastructure decisions from service-quality decisions. In particular, the variables $x_i$ determine which candidate nodes should be activated, whereas the variables $w_{ki}$ determine how the user demand is linked to the active backbone of the output solution. Given a transmission radius r, the communication graph contains an edge $(i,j)$ whenever the Euclidean distance between nodes i and j is at most r. The goal is to design a backbone that jointly accounts for connectivity and user service. To this end, two related paradigms are studied. The first is based on a minimum spanning tree (MST), which seeks a low-construction-cost connected backbone. The second is based on a dominating tree (DT), which additionally requires that every node belongs to the backbone or is adjacent to at least one active backbone node. In both cases, users are assigned to active nodes, which induces a trade-off between backbone-construction and user-assignment costs. This trade-off is controlled by the parameter $\alpha \in [0,1]$, which balances the relative importance of the two terms in the objective function.

The difference between the two models can be seen directly from a small geometric instance. In the MST-based model, the selected active nodes must be connected through a tree, but nodes that are not selected do not need to be adjacent to the backbone. As a consequence, the MST solution may concentrate the active nodes in a compact region when this reduces the weighted cost. The DT-based model adds an additional domination condition. Every node of the communication graph must either be active or have at least one active neighbor. Thus, the DT model can activate nodes that the MST model would leave out, especially when the graph is sparse and many vertices would otherwise remain far from the selected tree. This difference is illustrated in Figure 1 and Figure 2. These figures clearly illustrate how variations in the weighting parameter and the communication radius affect the resulting structures of the network. These examples motivate the need for different exact modeling approaches. Next, the sets, parameters, and decision variables that are common to all formulations are introduced.

• Sets
  • $V=\{1,\dots ,|V\left|\right\}$: set of candidate backbone nodes.
  • $A=\left\{\right(i,j):i,j\in V,i\ne j\}$: set of directed arcs of the communication graph.
  • $K=\{1,\dots ,|K\left|\right\}$: set of users.
• Parameters
  • $c_{ij}$: cost (or distance) associated with arc $(i,j)\in A$.
  • $d_{ki}$: assignment cost (or distance) between user $k\in K$ and node $i\in V$.
• Decision variables
  • $x_i\in \{0,1\}$: equals 1 if node i is selected as the active backbone node.
  • $z_{ij}\in \{0,1\}$: equals 1 if the arc $(i,j)\in A$ is selected in the backbone tree.
  • $w_{ki}\in \{0,1\}$: equals 1 if user $k\in K$ is assigned to node $i\in V$.

3.1. Miller–Tucker–Zemlin (MST and DT)

We first present Miller–Tucker–Zemlin (MTZ) formulations for the MST and DT models. These formulations enforce connectivity through ordering variables and provide a compact alternative to flow-based models, although they typically have weaker linear relaxations. For this reason, they are used mainly as benchmark formulations in our study.

$$\begin{array}{ccc}\hfill MS{T}_M:\underset{\{x,z,w,u\}}{min}& & \alpha \sum _{(i,j)\in A}{c}_{ij}{z}_{ij}+(1-\alpha )\sum _{k\in K}\sum _{i\in V}{d}_{ki}{w}_{ki}\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& & \sum _{(i,j)\in A}{z}_{ij}=\sum _{i\in V}{x}_i-1\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & \sum _{j:(j,i)\in A}{z}_{ji}\le x_i\forall i\in V\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & z_{ij}\le x_i\forall (i,j)\in A\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & z_{ij}\le x_j\forall (i,j)\in A\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & u_j-u_i-\left(\right|V|-1)z_{ij}-\left(\right|V|-3)z_{ji}\ge 2-\left|V\right|\forall (i,j)\in A\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & \sum _{i\in V}{w}_{ki}=1\forall k\in K\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & w_{ki}\le x_i\forall k\in K,i\in V\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & 1\le u_i\le \sum _{i\in V}{x}_i\forall i\in V\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & x\in {\{0,1\}}^{\left|V\right|},z\in {\{0,1\}}^{\left|A\right|},w\in {\{0,1\}}^{\left|K\right|\left|V\right|}\hfill \end{array}$$

(10)

The objective function (1) minimizes a weighted combination of the cost of backbone construction and the cost of user assignment. The constraint (2) imposes the tree cardinality condition on the set of active nodes. Constraints (3)–(5) link arc selection to node activation, ensuring that selected arcs can only connect active nodes. The constraint (6) corresponds to the classical MTZ ordering constraints and is used to eliminate disconnected cycles among the selected nodes through the auxiliary ordering variables u. The constraint (9) defines the domain of these variables. Finally, constraints (7) and (8) model user assignment by requiring each user to be assigned to exactly one active node, while constraint (10) defines the binary nature of the decision variables.

$$\begin{array}{c}{DT}_M:{\displaystyle \underset{\{x,z,w,u\}}{min}} \alpha {\displaystyle \sum _{(i,j)\in A}}{c}_{ij}{z}_{ij}+(1-\alpha ){\displaystyle \sum _{k\in K}}{\displaystyle \sum _{i\in V}}{d}_{ki}{w}_{ki}\\ \mathrm{s}.\mathrm{t}. x_j+{\displaystyle \sum _{i\in N\left(j\right)}}{x}_i\ge 1\forall j\in V\\ \left(2\right)–\left(10\right)\end{array}$$

(11)

The formulation DT_M extends MST_M by incorporating the domination constraints (11). These constraints require every node to be selected as an active backbone node or to be adjacent to at least one active backbone node. Therefore, DT_M preserves the tree, activation, and user-assignment structure of MST_M, while also enforcing the coverage of the nodes through domination.

3.2. Single-Flow Formulations (MST and DT)

To enforce connectivity through flow conservation, the set of directed arcs A is increased by introducing a fictitious root node r connected to every node in V. The resulting augmented arc set is

$$A_r=A\cup \{(r,j):j\in V\}.$$

The fictitious root acts only as a flow source and does not represent a physical network node. Based on this construction, the single-flow formulation for the MST model is given by

$$\begin{array}{c}MS{T}_F:{\displaystyle \underset{\{x,z,w,f\}}{min}} \alpha {\displaystyle \sum _{(i,j)\in A}}{c}_{ij}{z}_{ij}+(1-\alpha ){\displaystyle \sum _{k\in K}}{\displaystyle \sum _{i\in V}}{d}_{ki}{w}_{ki}\\ \mathrm{s}.\mathrm{t}. \left(2\right)–\left(5\right),\left(7\right)and\left(8\right)\\ {\displaystyle \sum _{j:(r,j)\in A_r}}{z}_{rj}=1\end{array}$$

(12)

$$\begin{array}{ccc}& & {\displaystyle \sum _{j:(r,j)\in A_r}}{f}_{rj}={\displaystyle \sum _{i\in V}}{x}_i\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & 0\le f_{ij}\le \left|V\right|z_{ij}\forall (i,j)\in A_r\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & {\displaystyle \sum _{j:(j,i)\in A_r}}{f}_{ji}-{\displaystyle \sum _{j:(i,j)\in A}}{f}_{ij}=x_i\forall i\in V\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & x\in {\{0,1\}}^{\left|V\right|}, z\in {\{0,1\}}^{|A_r|}, w\in {\{0,1\}}^{\left|K\right|\left|V\right|}, f\in {[0,\infty )}^{|A_r|}\hfill \end{array}$$

(16)

The formulation MST_F enforces connectivity through a single-commodity flow mechanism. Constraints inherited from the MTZ model preserve the tree cardinality condition, node–arc consistency, and user-assignment structure. The constraint (12) selects a unique root arc, while (13) injects an amount of flow equal to the number of active nodes. The constraint (14) links flow to arc selection, and (15) enforces flow conservation at each node, requiring that every active node absorbs one unit of flow. Together, these constraints guarantee that the selected active nodes are connected through a tree structure rooted at the fictitious source.

$$\begin{array}{c}D{T}_F:{\displaystyle \underset{\{x,z,w,f\}}{min}} \alpha {\displaystyle \sum _{(i,j)\in A}}{c}_{ij}{z}_{ij}+(1-\alpha ){\displaystyle \sum _{k\in K}}{\displaystyle \sum _{i\in V}}{d}_{ki}{w}_{ki}\\ \mathrm{s}.\mathrm{t}. \left(2\right)–\left(5\right),\left(7\right)and\left(8\right),\left(11\right)–\left(16\right)\end{array}$$

(17)

The formulation DT_F extends MST_F by adding the domination constraints (11). Hence, DT_F preserves the same flow-based connectivity mechanism and user-assignment structure as MST_F, while additionally requiring that every node be selected or adjacent to at least one active backbone node. The resulting solution is therefore a connected dominating tree.

3.3. Cut-Set Formulations

We finally consider cut-set formulations for the MST and DT models. In this family, connectivity is explicitly enforced through cut constraints defined on subsets of nodes. Although these formulations are often associated with strong linear relaxations, they involve exponentially many connectivity constraints and, therefore, require separation procedures in practice.

$$\begin{array}{c}MS{T}_C:{\displaystyle \underset{\{x,z,w\}}{min}} \alpha {\displaystyle \sum _{(i,j)\in A}}{c}_{ij}{z}_{ij}+(1-\alpha ){\displaystyle \sum _{k\in K}}{\displaystyle \sum _{i\in V}}{d}_{ki}{w}_{ki}\\ \mathrm{s}.\mathrm{t}. \left(2\right)–\left(5\right),\left(7\right)and\left(8\right),\left(10\right)\\ {\displaystyle \sum _{(i,j)\in \delta \left(S\right)}}{z}_{ij}\ge x_j\forall S\subset V,j\in S\end{array}$$

(18)

The formulation MST_C enforces connectivity by requiring that every subset of nodes containing an active node be linked to the remainder of the network through at least one selected arc. In this way, disconnected active components are excluded by cut-off constraints (18).

$$\begin{array}{ccc}\hfill D{T}_C:{\displaystyle {\displaystyle \underset{\{x,z,w\}}{min}}}& & \alpha {\displaystyle \sum _{(i,j)\in A}}{c}_{ij}{z}_{ij}+(1-\alpha ){\displaystyle \sum _{k\in K}}{\displaystyle \sum _{i\in V}}{d}_{ki}{w}_{ki}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& & \left(2\right)–\left(5\right),\left(7\right)and\left(8\right),\left(10\right),\left(11\right),\left(18\right)\hfill \end{array}$$

The formulation DT_C extends MST_C by adding the domination constraints (11). Therefore, DT_C maintains the same cut-set-based connectivity mechanism and user-assignment structure as MST_C, while also enforcing that every node is selected or adjacent to at least one active backbone node.

4. Theoretical Analysis, Valid Inequalities, and Algorithmic Components

In this section, the basic theoretical properties of the proposed models are first studied, including computational complexity and structural relations between the MST and DT formulations. Then, valid inequalities are introduced for the single-flow model, which provide the best balance between formulation strength and computational tractability in our experiments. Finally, two algorithmic components are presented, built on this formulation: a lazy-cut separation procedure for the exponential connectivity cuts and a local branching matheuristic for larger instances [27].

4.1. Complexity Results and Structural Relations

Although the proposed formulations are inspired by spanning tree and dominating tree structures, the optimization models studied in this paper are more general than their classical counterparts because they jointly determine node activation, backbone connectivity, and user assignment. As a result, both models inherit the combinatorial difficulty of well-known hard network design problems. The following results formalize this fact.

Theorem 1.

The MST-based model studied in this paper is NP-hard for any $\alpha \in (0,1)$.

Proof. 

We prove the result by reduction from the Steiner tree problem (STP), which is NP-hard. Consider an instance of the STP defined on an undirected graph $G=(V,E)$ with edge costs ${\tilde{c}}_{ij}\ge 0$ and a set of terminals $R\subseteq V$. An instance of the MST-based model is constructed as follows. Let the node set be V, and define the directed arc set as the bi-directed version of E, that is,

$$A=\left\{\right(i,j),(j,i):\{i,j\}\in E\}.$$

For each $(i,j)\in A$, set $c_{ij}={\tilde{c}}_{ij}$. Create one user for each terminal; that is, let $K=R$. For each $k\in K$, define the assignment costs

$$d_{ki}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}i=k,\hfill \\ M,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

where $M>0$ is a sufficiently large constant such that

$$(1-\alpha )M>\alpha {\displaystyle \sum _{\{i,j\}\in E}}{\tilde{c}}_{ij}.$$

Consider the objective function

$$\alpha {\displaystyle \sum _{(i,j)\in A}}{c}_{ij}{z}_{ij}+(1-\alpha ){\displaystyle \sum _{k\in K}}{\displaystyle \sum _{i\in V}}{d}_{ki}{w}_{ki},\alpha \in (0,1).$$

Because assigning a user k to any node $i\ne k$ incurs a penalty M, any optimal solution must assign each user k to node k and therefore activate all nodes in R. Hence, all terminals must be included in the set of active nodes. The tree constraints enforce that the selected arcs form a connected tree over the active nodes. Additional non-terminal nodes may be activated only if they help reduce the total infrastructure cost. Therefore, once all terminals are forced to be active, the problem reduces to finding a minimum-cost tree connecting the terminals, possibly using additional non-terminal nodes. This is exactly the Steiner tree problem. Since the Steiner tree problem is NP-hard, the MST-based model is NP-hard.    □

Corollary 1.

For $\alpha =1$, the problem reduces to a classical minimum spanning tree problem on active nodes and can be solved in polynomial time.

Corollary 2.

For $\alpha =0$, the problem reduces to a coverage-oriented network design problem that remains combinatorial, as it captures elements of both the location of the facility and the dominant structures.

The computational complexity therefore arises from the interaction between connectivity and coverage, rather than from either component in isolation.

Theorem 2.

The DT-based model studied in this paper is NP-hard.

Proof. 

We prove the result by reduction from the Minimum Connected Dominating Set Problem, which is NP-hard. Consider an undirected graph $G=(V,E)$. Construct the corresponding DT instance on the same node set V and on the bi-directed arc set

$$A=\left\{\right(i,j),(j,i):\{i,j\}\in E\}.$$

Set

$$\alpha =1,c_{ij}=1\forall (i,j)\in A.$$

Since $\alpha =1$, the user-assignment term disappears from the objective function, and the model reduces to

$$min{\displaystyle \sum _{(i,j)\in A}}{z}_{ij}.$$

By the DT constraints, the selected active nodes must satisfy two properties: (i) they form a dominating set since every node must be selected or adjacent to a selected node; and (ii) they induce a connected tree. Moreover, the tree cardinality condition is enforced

$${\displaystyle \sum _{(i,j)\in A}}{z}_{ij}={\displaystyle \sum _{i\in V}}{x}_i-1.$$

Since every selected arc has a unit cost, minimizing the objective is equivalent to minimizing

$${\displaystyle \sum _{i\in V}}{x}_i-1,$$

that is, minimizing the number of active nodes. Therefore, the DT-based model is exactly looking for a dominating set connected to minimum cardinality. This is precisely the Minimum Connected Dominating Set Problem. Since this problem is NP-hard, the DT-based model studied in this paper is NP-hard.    □

This remains true even for $\alpha =1$, and therefore also for the general case.

Remark 1.

The previous results show that the difficulty of the proposed models does not come from the classical optimization of the spanning tree itself, but from the interaction between connectivity, node activation, and coverage requirements. In particular, the MST-based model becomes difficult because user-assignment decisions implicitly force terminal selection, while the DT-based model remains hard even when the user-assignment term is removed due to the connected domination structure.

Proposition 1.

Let $z_{MS{T}_F}^{★}$ and $z_{D{T}_F}^{★}$ denote the optimal objective values of models MST_F and DT_F, respectively, defined on the same directed graph $G=(V,A)$, with the same parameter $\alpha \in [0,1]$, the same user set K, and the same objective coefficients. Then

$$z_{MS{T}_F}^{★}\le z_{D{T}_F}^{★}.$$

Hence, the dominating tree formulation provides an upper bound on the optimal value of the spanning tree formulation.

Proof. 

We compare the feasible regions of both models. The formulation MST_F is defined by the same constraints as DT_F, except that DT_F includes the additional domination constraints. Both models share the same objective function and the same structural, flow, and assignment constraints. Therefore, every feasible solution of DT_F is also feasible for MST_F. If ${\mathcal{F}}_{MST}$ and ${\mathcal{F}}_{DT}$ denote the feasible sets of MST_F and DT_F, respectively, then

$${\mathcal{F}}_{DT}\subseteq {\mathcal{F}}_{MST}.$$

Since both models are minimization problems with the same objective function, minimizing the smaller feasible set ${\mathcal{F}}_{DT}$ cannot yield a value smaller than minimizing the larger feasible set ${\mathcal{F}}_{MST}$. Hence,

$$z_{MS{T}_F}^{★}=min\{f(x,z,w,f):(x,z,w,f)\in {\mathcal{F}}_{MST}\}\le$$

$$min\{f(x,z,w,f):(x,z,w,f)\in {\mathcal{F}}_{DT}\}=z_{D{T}_F}^{★}.$$

Thus, the optimal value of DT_F is an upper bound on the optimal value of MST_F.    □

4.2. Structural Convergence Hypothesis

The previous results establish a basic dominance relation between the feasible regions and the objective values of the MST and DT formulations. A natural question is whether, under certain operating regimes, the domination constraints become effectively redundant. This allows us to write the following empirical hypothesis.

Hypothesis 1.

In the $(\alpha ,r)$ parameter space, MST- and DT-based solutions are expected to become structurally similar when the communication radius r is sufficiently large and the weighting parameter α assigns enough importance to user-assignment costs. In such regimes, the optimal or near-optimal MST backbone may already satisfy, or nearly satisfy, the domination requirement. As such, the additional domination constraints for DT may become weakly binding or practically irrelevant.

This statement should be classified as an empirical structural hypothesis instead of a theoretical guarantee. The intuition is that decreasing $\alpha$ gives more importance to the user-assignment cost, allowing for the activation of spatially distributed nodes, while increasing r makes the communication graph denser and increases the likelihood that active nodes dominate the remaining vertices. In the computational section, this hypothesis is examined in more depth by considering the simultaneous effect of both $\alpha$ and r on objective values, solution structures, and collapse regions between MST and DT formulations.

4.3. Valid Inequalities for the Single-Flow Formulation

Since solving the models poses a challenging fact, strengthening the formulations emerges as an essential strategy to improve computational performance. Our preliminary experiments show that the single-flow formulation provides the best balance between scalability and solution quality, particularly for the larger ones. For this reason, the valid inequalities focus on the flow-based model. Although similar ideas could be extended to other formulations, the single-flow model is the most suitable setting due to its empirical strength and ability to accommodate additional cuts.

In concrete, the valid inequalities proposed exploit the interaction between node activation, connectivity, and flow conservation. Two types of cuts are considered: static and exponential inequalities. The former can be added from the outset, whereas the latter are better handled through dynamic separation within a branch-and-cut framework [11]. Let us $A_r=A\cup \{(r,i):i\in V\}$, where r is the fictitious root. For any subset $S\subseteq V$, define

$$A\left(S\right):=\{(i,j)\in A:i\in S,j\in S\},{\delta }^{-}\left(S\right):=\{(i,j)\in A:i\notin S,j\in S\}$$

• Family 1: Root-activation inequalities.

For every node $i\in V$,

$$z_{ri}\le x_i\forall i\in V.$$

(19)

Proposition 2.

Inequalities (19) are valid for single-flow MST and DT formulations.

Proof. 

If $z_{ri}=1$, then the fictitious root sends the flow directly to node i through the arc $(r,i)$. In the single-flow model, flow is injected only to support the active nodes, and each active node absorbs one unit of flow. Therefore, any node incident to a selected root arc must belong to the active backbone, which implies $x_i=1$. Hence, (19) is valid.    □

• Family 2: Incident-support inequalities.

For every node $i\in V$,

$$x_i\le z_{ri}+{\displaystyle \sum _{j:(j,i)\in A}}{z}_{ji}+{\displaystyle \sum _{j:(i,j)\in A}}{z}_{ij}\forall i\in V.$$

(20)

Proposition 3.

Inequalities (20) are valid for single-flow MST and DT formulations.

Proof. 

If $x_i=0$, then (20) holds trivially. Assume now that $x_i=1$. Since active nodes must induce a connected backbone, node i must be incident to the selected structure. More precisely, either (i) the node i is the unique node adjacent to the fictitious root, in which case $z_{ri}=1$; or (ii) some selected structural arc enters i; or (iii) some selected structural arc leaves i. If none of these situations occur, then node i would be active but isolated from the backbone, contradicting connectivity. Therefore, the right-hand side of (20) must be at least one whenever $x_i=1$, which proves the validity.    □

• Family 3: Root-aware connectivity cuts.

For every nonempty subset $S\subseteq V$ and every node $h\in S$,

$${\displaystyle \sum _{(i,j)\in {\delta }^{-}\left(S\right)}}{z}_{ij}+{\displaystyle \sum _{j\in S}}{z}_{rj}\ge x_h.$$

(21)

Proposition 4.

Inequalities (21) are valid for single-flow MST and DT formulations.

Proof. 

Fix a nonempty subset $S\subseteq V$ and a node $h\in S$. If $x_h=0$, then (21) is trivially satisfied. Assume $x_h=1$. Since node h is active, it must belong to the unique connected component of the selected backbone. Therefore, the subset S that contains h must be connected to the root-supported backbone in one of the following two ways: (i) some selected structural arc enters S from $V\setminus S$, that is,

$${\displaystyle \sum _{(i,j)\in {\delta }^{-}\left(S\right)}}{z}_{ij}\ge 1;$$

or (ii) the unique selected root arc enters a node of S, that is,

$${\displaystyle \sum _{j\in S}}{z}_{rj}\ge 1.$$

If both terms on the left-hand side were zero, then no selected arc from $V\setminus S$ would enter S, and no selected root arc would enter S. Hence, the active node h could not be reached from the root-supported component, contradicting the feasibility of the single-flow formulation. Therefore, (21) is valid.    □

Remark 2.

Inequalities (19) and (20) are static and can be added directly to the model. In contrast, the family (21) is exponential in the number of subsets $S\subseteq V$ and is therefore better handled through dynamic separation within a lazy-cut framework.

4.4. Separation Procedure and Matheuristic Framework

In addition to the exact formulations and valid inequalities presented above, two complementary algorithmic components for the single-flow model are considered. First, a lazy-cut separation procedure is incorporated to dynamically enforce the exponential family of root-aware connectivity inequalities only when violated by incumbent solutions. This avoids introducing all such constraints explicitly while preserving exactness. Second, motivated by the strong computational behavior of the single-flow formulation, a local branching matheuristic in the spirit of [27] is designed to intensify the search for high-quality incumbent solutions and improve scalability for larger instances.

The lazy-cut component strengthens the exact branch-and-cut procedure by eliminating disconnected incumbent solutions. In turn, the local branching approach allows restricting the search to a neighborhood around a reference solution through a Hamming-distance-type constraint on the node activation variables. The latter enables a focused exploration of promising regions of the solution space. From an algorithmic point of view, notice that the lazy-cut procedure preserves exactness while avoiding the explicit enumeration of exponentially many connectivity inequalities. The local branching approach is utilized as a scalable matheuristic for large or difficult instances in which full optimality certification becomes computationally prohibitive.

The separation procedure of Algorithm 1 does not enumerate the exponentially many inequalities. Instead of that, it checks only whether the current incumbent integer solution contains active components that are disconnected from the root-supported component. This can be performed by building the selected digraph induced by the arcs with ${\overline{z}}_{ij}=1$ and performing a breadth-first or depth-first search from the unique node selected by the artificial root. Hence, each separation call requires $O\left(\right|V|+|\overline{A}\left|\right)$ time, where $\overline{A}=\{(i,j)\in A:{\overline{z}}_{ij}=1\}$. Since $|\overline{A}|\le |A|$, the worst-case complexity of one separation call is $O\left(\right|V|+|A\left|\right)$. Therefore, although the family of cuts is exponential, the implemented separation oracle is polynomial and computationally inexpensive compared with solving the underlying mixed-integer program.

The local branch constraint restricts the search to solutions whose activation pattern differs from the incumbent in at most k binary decisions, thus defining a compact and meaningful neighborhood around the current backbone structure. The scalability of Algorithm 2 comes from the fact that the local branching constraint is imposed on the activation vector x, which represents the main strategic layer of the problem. Once a neighborhood around the incumbent activation pattern is fixed, the remaining connectivity, flow, and assignment variables are optimized within a restricted but still meaningful subproblem. The parameter $\varphi$ is used to control the size of the neighborhood: small values intensify the search around the incumbent. Larger values, on the other hand, allow for diversification when no improvement is obtained. By doing so, the method avoids exploring the full binary space at each iteration and instead solves a sequence of smaller, focused mixed-integer subproblems.

The following result establishes the correctness of the lazy-cut separation procedure associated with inequalities (21).

Theorem 3.

The lazy-cut separation procedure for inequalities (21) is valid and does not remove any feasible integer solution from the single-flow formulation.

Proof. 

The family of inequalities (21) is valid for the formulation, as shown above. Therefore, any feasible integer solution must satisfy all such inequalities. The separation procedure only adds cuts when a candidate integer solution $(\overline{x},\overline{z})$ violates at least one inequality from the family. In particular, if there exists a subset $C\subseteq V$ containing at least one active node that is not reachable from the root-supported component, then

$${\displaystyle \sum _{(i,j)\in {\delta }^{-}\left(C\right)}}{\overline{z}}_{ij}+{\displaystyle \sum _{j\in C}}{\overline{z}}_{rj}=0,$$

violates (21). □

The separation procedure runs in linear time and is therefore computationally negligible compared to solving the underlying mixed-integer program.

In summary, the lazy-cut procedure is used as an exact branch-and-cut algorithm because it separates only valid connectivity inequalities and never removes feasible connected solutions. In contrast, the local branching procedure is used as a matheuristic component because it sacrifices full optimality certification in favor of faster exploration of high-quality neighborhoods. These two components, therefore, play complementary roles: the former improves exactness and formulation enforcement, whereas the latter enhances practical scalability for the largest and most challenging instances.

5. Computational Experiments

The computational experiment is designed to ensure reproducibility and fair comparisons across the proposed models. For each instance, the node coordinates are generated uniformly at random in the unit square ${[0,1]}^2$. A fixed random seed is used for each instance size, and the same seeds are used for all MST and DT formulations. The cost matrices C and D are calculated using the coordinates of the nodes to compute the Euclidean distances between the nodes and between the users and the nodes, respectively. To ensure a fair comparison, all models are solved on the same input graphs. The exact approaches are solved on instances with up to 100 nodes, both with and without valid inequalities, while the local branching approach is applied to instances with up to 150 nodes.

In general, the number of users is set to $\left|K\right|=5N$, unless otherwise stated. This value $\left|K\right|=5N$ is used to represent a demand that is denser than the candidate backbone. Notice that this is commonly done in communication settings where a limited number of infrastructure nodes must serve a larger number of users. The solver time limit is arbitrarily fixed at 3600 s, and all reported objective function values, CPU times in seconds, number of branch-and-bound nodes, and optimality gaps correspond to the same computational experiments for all the instances. The communication graph is constructed by connecting two nodes if their Euclidean distance is at most the radius r. Only connected graphs are saved for the experiments. The weighting parameter is varied in $\alpha \in \{0.25,0.50,0.75,0.95\}$, and the baseline communication radius is set to $r=0.20$, unless another specific sensitivity analysis with respect to r is mentioned. The number of lazy constraints is also reported. For the joint sensitivity analysis in the $(\alpha ,r)$ space, the same instance-generation procedure is utilized, and both parameters are varied over a grid of values. This design allows us to isolate the effect of $\alpha$ and r on the structural similarity between MST and DT solutions since all other data-generation components remain fixed across the compared models. In summary, this section measures the performances of the proposed formulations and algorithmic enhancements through a comprehensive computational study. The analysis is organized according to the following questions: (i) how the different exact formulations compare in terms of solution quality and computational effort, (ii) how much the proposed valid inequalities strengthen the single-flow models, (iii) whether the local branching matheuristic improves performance on the hardest instances, and (iv) under which parameter regimes MST and DT solutions become structurally similar or clearly different. All the experiments were conducted on a machine using Windows 11 Pro Version 25H2, an Intel Core i7-12700H processor (2.30 GHz), and 16 GB of RAM. In addition, all models were implemented in Python 3.13.9 and solved with the Gurobi solver [11].

5.1. Comparison of Exact Formulations

Initially, the MTZ, single-flow, and cut-set models are compared for both MST and DT. For each model, the objective value, the number of branch-and-bound nodes explored (B&B), the CPU time in seconds, and the optimality gaps are reported. For the cut-set formulations, the number of dynamically generated lazy connectivity cuts is also present. These metrics allow us to assess both formulation strength and computational scalability.

Table 2. Comparison between MST and dominating tree MTZ formulations using $\alpha =0.25$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
10	7.35	1	0.02	0.00	7.35	1	0.02	0.00
15	8.38	632	0.39	0.00	8.38	873	0.38	0.00
20	9.47	1	0.51	0.00	9.47	1	0.30	0.00
25	12.61	650,731	56.49	0.00	12.61	679,249	55.53	0.00
30	12.06	296,611	106.07	0.00	12.06	1,290,107	619.64	0.00
35	12.36	8,110,645	3600.14	0.68	12.36	7,711,193	3600.14	0.65
40	13.62	3,455,301	1289.35	0.00	13.62	2,308,009	726.51	0.00
45	16.10	3,580,234	3600.29	0.76	16.10	3,858,949	3600.42	0.76
50	14.38	4,027,043	3600.62	0.66	14.38	4,457,466	3600.27	0.67

,

Table 3. Comparison between MST and dominating tree MTZ formulations using $\alpha =0.5$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
10	6.14	1	0.03	0.00	6.26	1	0.02	0.00
15	6.69	5304	0.82	0.00	6.69	1492	0.30	0.00
20	7.55	178	1.06	0.00	7.55	41	0.93	0.00
25	9.41	1,237,964	593.18	0.00	9.41	723,656	374.42	0.00
30	9.92	586,156	709.68	0.00	9.92	104,400	58.75	0.00
35	9.65	1,989,306	3600.36	1.81	9.65	1,720,760	3600.43	1.94
40	10.56	1,849,591	3600.16	0.56	10.56	1,764,942	3600.33	0.67
45	12.36	1,700,968	3600.27	2.18	12.36	1,476,840	3600.43	2.11
50	11.25	680,313	3600.54	1.92	11.25	735,606	3600.53	1.88

,

Table 4. Comparison between MST and dominating tree MTZ formulations using $\alpha =0.75$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
10	3.99	1	0.03	0.00	5.13	1	0.03	0.00
15	4.75	4555	0.71	0.00	4.83	680	0.32	0.00
20	5.31	5576	1.31	0.00	5.41	925	0.63	0.00
25	6.05	909,274	1107.13	0.00	6.05	1,342,750	1743.81	0.00
30	7.26	1,276,124	3600.26	1.51	7.44	2,953,830	3600.34	5.49
35	6.82	1,201,266	3600.33	4.89	6.84	1,111,271	3600.70	5.22
40	7.39	951,022	3600.90	2.84	7.39	829,905	3600.53	2.58
45	8.44	676,890	3601.20	6.11	8.46	686,970	3601.11	6.31
50	7.82	531,311	3601.13	3.76	7.82	519,348	3601.03	3.47

and

Table 5. Comparison between MST and dominating tree MTZ formulations using $\alpha =0.95$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
10	1.06	1	0.03	0.00	4.22	1	0.03	0.00
15	1.60	1297	0.53	0.00	3.03	733	0.34	0.00
20	1.84	1433	0.77	0.00	3.38	1	0.54	0.00
25	2.08	36,506	27.38	0.00	2.83	481,430	330.80	0.00
30	2.59	156,406	181.02	0.00	4.54	277,233	240.89	0.00
35	3.04	1,397,563	3600.59	9.96	3.70	1,468,043	3601.08	8.73
40	3.16	631,244	3600.76	14.74	3.58	888,550	3600.65	7.82
45	3.21	482,032	3600.30	9.53	3.97	544,442	3600.33	18.08
50	3.43	340,909	3600.43	12.14	3.83	486,383	3600.81	11.15

report the performance of MTZ formulations for different values of $\alpha$ with a fixed radius $r=0.2$. From a structural point of view, MST and DT typically produce identical or very similar objective values, especially for small and moderate values of $\alpha$. However, the computational behavior of the MTZ models deteriorates rapidly as the size of the problem increases. Both the number of branch-and-bound nodes and the CPU time grow sharply, and several medium- and large-scale instances reach the time limit with nonzero optimality gaps. This confirms that, although MTZ formulations are compact, they do not provide sufficiently strong relaxations for the larger instances considered here.

Table 6. Comparison between MST and dominating tree flow formulations using $\alpha =0.25$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
10	7.35	1.0	0.03	0.00	7.35	1.0	0.03	0.00
15	8.38	1.0	0.12	0.00	8.38	1.0	0.10	0.00
20	9.47	1.0	0.12	0.00	9.47	1.0	0.10	0.00
25	12.61	1.0	0.18	0.00	12.61	1.0	0.21	0.00
30	12.06	1.0	0.37	0.00	12.06	1.0	0.22	0.00
35	12.36	1.0	0.78	0.00	12.36	106.0	0.47	0.00
40	13.62	1.0	0.82	0.00	13.62	1.0	0.47	0.00
45	16.10	1.0	0.50	0.00	16.10	1.0	0.47	0.00
50	14.38	2378.0	1.56	0.00	14.38	4336.0	1.10	0.00
55	16.34	2523.0	1.39	0.00	16.34	423.0	1.98	0.00
60	16.96	169.0	5.40	0.00	16.96	3601.0	3.78	0.00
65	17.36	1.0	6.36	0.00	17.36	4291.0	5.19	0.00
70	17.81	550.0	5.97	0.00	17.81	1096.0	8.04	0.00
75	17.49	7478.0	8.91	0.00	17.49	4301.0	4.99	0.00
80	19.43	11,654.0	9.43	0.00	19.43	6554.0	8.21	0.00
85	19.88	23,361.0	24.56	0.00	19.88	2593.0	14.79	0.00
90	21.03	3249.0	11.50	0.00	21.03	3284.0	12.01	0.00
95	21.24	19,950.0	15.70	0.00	21.24	3775.0	11.52	0.00
100	22.05	3103.0	14.86	0.00	22.05	3357.0	15.04	0.00

,

Table 7. Comparison between MST and dominating tree flow formulations using $\alpha =0.5$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
10	6.14	1.0	0.04	0.00	6.26	1.0	0.03	0.00
15	6.69	1.0	0.24	0.00	6.69	1.0	0.12	0.00
20	7.55	1.0	0.25	0.00	7.55	1.0	0.21	0.00
25	9.41	1.0	0.48	0.00	9.41	1.0	0.30	0.00
30	9.92	1.0	0.52	0.00	9.92	1.0	0.39	0.00
35	9.65	1.0	1.88	0.00	9.65	1.0	0.43	0.00
40	10.56	1.0	1.54	0.00	10.56	1.0	0.73	0.00
45	12.36	1.0	1.68	0.00	12.36	1.0	1.05	0.00
50	11.25	1.0	3.19	0.00	11.25	1.0	1.89	0.00
55	12.82	1.0	3.98	0.00	12.82	1719.0	1.75	0.00
60	13.20	1.0	5.74	0.00	13.20	654.0	2.48	0.00
65	13.27	4043.0	10.23	0.00	13.27	4100.0	9.11	0.00
70	13.98	1.0	24.46	0.00	13.98	1.0	12.46	0.00
75	13.54	3627.0	9.09	0.00	13.54	9833.0	8.86	0.00
80	14.81	2435.0	14.88	0.00	14.81	4498.0	23.49	0.00
85	15.26	6871.0	27.99	0.00	15.26	3229.0	42.17	0.00
90	16.33	5561.0	36.28	0.00	16.33	3270.0	45.46	0.00
95	16.27	3007.0	44.76	0.00	16.27	10,852.0	42.25	0.00
100	16.82	4198.0	52.63	0.00	16.82	9620.0	54.20	0.00

,

Table 8. Comparison between MST and dominating tree flow formulations using $\alpha =0.75$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
10	3.99	1.0	0.05	0.00	5.13	1.0	0.03	0.00
15	4.75	1.0	0.33	0.00	4.83	1.0	0.23	0.00
20	5.31	53.0	1.06	0.00	5.41	1.0	0.25	0.00
25	6.05	1636.0	3.13	0.00	6.05	1.0	0.77	0.00
30	7.26	1.0	3.29	0.00	7.44	29.0	3.67	0.00
35	6.82	1752.0	6.37	0.00	6.84	328.0	1.20	0.00
40	7.39	1263.0	10.11	0.00	7.39	1.0	1.95	0.00
45	8.44	3003.0	30.21	0.00	8.44	4194.0	26.78	0.00
50	7.82	2786.0	62.59	0.00	7.82	2707.0	20.29	0.00
55	8.56	13,868.0	77.50	0.00	9.14	2921.0	15.22	0.00
60	9.18	2753.0	107.61	0.00	9.18	2584.0	31.75	0.00
65	8.97	5018.0	185.50	0.00	8.97	9622.0	142.21	0.00
70	9.94	2598.0	207.23	0.00	9.94	3163.0	88.87	0.00
75	9.31	2521.0	123.05	0.00	9.31	2366.0	58.79	0.00
80	9.94	3268.0	238.75	0.00	9.94	10,222.0	264.94	0.00
85	10.29	27,903.0	619.84	0.00	10.29	35,403.0	712.72	0.00
90	11.18	13,083.0	787.26	0.00	11.36	3270.0	482.26	0.00
95	11.02	2489.0	518.75	0.00	11.02	2479.0	563.81	0.00
100	11.34	39,039.0	1901.19	0.00	11.34	31,457.0	1604.26	0.00

and

Table 9. Comparison between MST and dominating tree flow formulations using $\alpha =0.95$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
10	1.06	1.0	0.08	0.00	4.22	1.0	0.03	0.00
15	1.60	75.0	0.35	0.00	3.03	1.0	0.13	0.00
20	1.84	174.0	2.44	0.00	3.38	1.0	0.47	0.00
25	2.08	2485.0	6.10	0.00	2.83	1.0	3.03	0.00
30	2.59	2260.0	13.52	0.00	4.54	1.0	2.23	0.00
35	3.04	3441.0	22.93	0.00	3.70	99.0	2.08	0.00
40	3.16	9965.0	59.77	0.00	3.58	1.0	2.73	0.00
45	3.21	36,494.0	383.58	0.00	3.97	6818.0	90.97	0.00
50	3.43	58,426.0	1593.83	0.00	3.83	12,135.0	108.96	0.00
55	3.53	38,874.0	3600.45	5.68	4.81	3719.0	53.67	0.00
60	3.91	156,541.0	3600.72	6.16	4.43	48,655.0	1227.12	0.00
65	3.86	39,073.0	3600.62	11.96	4.13	80,663.0	3416.43	0.00
70	4.43	59,629.0	3600.74	4.65	4.91	40,342.0	1060.46	0.00

report the performance of the single-flow formulations. In contrast to MTZ models, flow-based formulations exhibit substantially better computational behavior. Most instances are solved to optimality with zero gap, and the number of explored nodes remains comparatively small even as the instance size increases. This suggests that the single-flow model provides a much tighter relaxation and a better trade-off between formulation size and strength. As in the MTZ case, MST and DT often produce identical or nearly identical objective values for a broad range of instances, which already provides computational evidence of their structural proximity in some operating regimes. Although neither MST_F nor DT_F uniformly dominates the other computationally, the general message is clear: among the exact models tested, the single-flow formulation is the most scalable and robust.

Table 10. Comparison between MST and dominating tree cut-set formulations using $\alpha =0.25$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	MST Lazy	DT Obj	DT B&B	CPU (s)	Gap (%)	DT Lazy
10	7.35	1	0.03	0.00	3	7.35	1	0.02	0.00	3
15	8.38	1	0.13	0.00	37	8.38	97	0.11	0.00	51
20	9.47	47	0.23	0.00	59	9.47	186	0.23	0.00	103
25	12.61	13,050	7.36	0.00	909	12.61	10,777	5.96	0.00	870
30	12.06	10,399	7.24	0.00	1399	12.06	13,371	9.18	0.00	1848
35	12.36	19,607	24.73	0.00	3053	12.36	75,197	261.81	0.00	8424
40	13.62	124,060	809.00	0.00	11,271	13.62	113,155	752.20	0.00	14,454
45	65.64	184,501	3603.69	75.94	24,277	-	188,753	3602.77	-	22,511
50	44.07	173,179	3603.97	68.15	18,426	-	192,631	3602.93	-	17,729

-: No solution found in 1 h of CPU time.

,

Table 11. Comparison between MST and dominating tree cut-set formulations using $\alpha =0.5$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	MST Lazy	DT Obj	DT B&B	CPU (s)	Gap (%)	DT Lazy
10	6.14	1	0.04	0.00	4	6.26	1	0.03	0.00	3
15	6.69	1	0.33	0.00	62	6.81	232	0.19	0.00	86
20	7.57	334	0.35	0.00	117	7.56	1000	0.35	0.00	315
25	9.44	12,200	10.37	0.00	1293	9.50	22,970	11.83	0.00	1686
30	9.92	15,829	13.56	0.00	2568	9.92	11,994	10.01	0.00	2738
35	9.66	93,945	415.63	0.00	10,326	9.65	211,226	2216.36	0.00	20,930
40	10.56	179,480	1896.39	0.00	13,623	-	158,361	3604.53	-	27,529
45	52.82	172,393	3603.04	77.77	23,914	-	170,702	3603.10	-	22,556
50	17.59	164,457	3603.34	39.91	19,080	-	167,563	3603.99	-	19,853

-: No solution found in 1 h of CPU time.

,

Table 12. Comparison between MST and dominating tree cut-set formulations using $\alpha =0.75$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	MST Lazy	DT Obj	DT B&B	CPU (s)	Gap (%)	DT Lazy
10	3.99	3	0.16	0.00	5	5.13	1	0.03	0.00	3
15	4.85	363	0.32	0.00	92	4.83	231	0.25	0.00	112
20	5.47	4544	1.30	0.00	1205	5.49	4016	0.83	0.00	801
25	6.06	82,843	141.98	0.00	5295	6.26	94,344	190.75	0.00	5766
30	7.32	197,059	2055.33	0.00	21,592	7.65	147,473	1974.86	0.00	23,660
35	7.18	160,647	3605.48	9.27	34,683	-	149,876	3607.57	-	32,457
40	8.04	137,025	3603.61	15.70	29,134	-	131,182	3603.46	-	33,462
45	12.99	173,338	3603.62	42.67	19,494	-	170,585	3603.17	-	19,322
50	20.27	162,201	3603.42	65.72	19,959	-	150,598	3603.69	-	23,212

-: No solution found in 1 h of CPU time.

and

Table 13. Comparison between MST and dominating tree cut-set formulations using $\alpha =0.95$ and number of users $5N$.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	MST Lazy	DT Obj	DT B&B	CPU (s)	Gap (%)	DT Lazy
10	1.06	1	0.17	0.00	1	4.22	1	0.03	0.00	5
15	1.60	117	0.38	0.00	28	3.03	605	0.34	0.00	253
20	1.87	252	1.26	0.00	48	3.47	7823	2.91	0.00	2048
25	2.19	38,996	57.74	0.00	4436	3.07	318,982	2772.20	0.00	22,241
30	2.88	79,269	397.22	0.00	9829	4.62	148,463	3606.68	26.77	34,103
35	3.17	98,263	913.51	0.00	15382	-	148,696	3605.86	-	29,871
40	3.26	197,609	3612.64	4.13	25104	-	148,531	3604.40	-	31,406
45	3.34	193,890	3601.38	13.99	13732	-	147,619	3603.49	-	21,376
50	4.01	166,874	3603.05	28.00	9619	-	157,008	3602.72	-	13,219

-: No solution found in 1 h of CPU time.

report the performance of cut-set formulations, including lazy constraint separation. From a modeling perspective, these formulations provide an explicit connectivity representation and are often associated with strong relaxations. In practice, however, they are significantly more expensive than the single-flow model in this setting. As instance size increases, both MST and DT cut-set formulations show rapid growth in branch-and-bound nodes, CPU time, and the number of lazy cuts. In several large instances, the solver reaches the time limit or fails to find a feasible solution within one hour. The objective values remain broadly consistent with those of the other formulations, but the computational burden is clearly much higher.

Taken together, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12 and Table 13 lead to two main conclusions. First, MST and DT frequently produce very similar objective values, especially in coverage-oriented or denser regimes. Second, from a computational perspective, the single-flow formulation is the most effective exact approach, clearly outperforming MTZ and cut-set models in terms of tractability and scalability. For this reason, the remainder of the computational analysis focuses on the flow-based formulation.

This behavior suggests that, for the geometric backbone design problems studied here, the strength of the cut-set representation is offset by the cost of dynamic separation and by the large number of cuts required in practice.

5.2. Impact of Valid Inequalities

Next, the impact of the proposed valid inequalities on the single-flow formulation is evaluated. In this part of the study, both the strengthened and unstrengthened flow models are compared in terms of solution quality and computational effort. Since the exponential family of cuts proved computationally expensive and offered limited practical benefits, only the static valid inequalities are considered hereafter.

Table 14. Comparison between MST and dominating tree flow formulations using $\alpha =0.25$, $r=0.2$ and number of users $5N$ using valid inequalities.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
70	17.81	24	3.28	0.00	17.81	1	2.38	0.00
75	17.49	1	2.02	0.00	17.49	163	1.63	0.00
80	19.43	1	3.30	0.00	19.43	75	3.20	0.00
85	19.88	490	4.65	0.00	19.88	1	4.43	0.00
90	21.03	1	4.50	0.00	21.03	1	4.02	0.00
95	21.24	1	4.86	0.00	21.24	1	4.72	0.00
100	22.05	1	5.83	0.00	22.05	1	6.82	0.00
120	23.47	22	11.04	0.00	23.47	369	10.89	0.00
140	25.75	773	17.61	0.00	25.75	299	17.65	0.00
150	26.71	1	20.42	0.00	26.71	27	19.66	0.00

,

Table 15. Comparison between MST and dominating tree flow formulations using $\alpha =0.5$, $r=0.2$ and number of users $5N$ using valid inequalities.

N	MST Obj	MST B&B	CPU(s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
70	13.98	1	8.34	0.00	13.98	1	10.67	0.00
75	13.54	1	5.24	0.00	13.54	1	2.96	0.00
80	14.81	933	10.81	0.00	14.81	933	12.36	0.00
85	15.26	868	20.76	0.00	15.26	6055	19.96	0.00
90	16.33	1	15.23	0.00	16.33	1	15.54	0.00
95	16.27	30	15.52	0.00	16.27	433	18.74	0.00
100	16.82	345	20.58	0.00	16.82	1210	23.39	0.00
120	17.99	79	26.48	0.00	17.99	1	24.88	0.00
140	19.70	3117	78.79	0.00	19.70	2474	72.38	0.00
150	20.42	4079	80.47	0.00	20.42	8652	84.11	0.00

,

Table 16. Comparison between MST and dominating tree flow formulations using $\alpha =0.75$, $r=0.2$ and number of users $5N$ using valid inequalities.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
70	9.94	3355	152.63	0.00	9.94	2556	129.36	0.00
75	9.31	2397	109.42	0.00	9.31	2340	44.21	0.00
80	9.94	12,301	303.96	0.00	9.94	2357	411.79	0.00
85	10.29	25,036	395.19	0.00	10.29	9627	347.78	0.00
90	11.18	3694	498.73	0.00	11.36	3532	426.21	0.00
95	11.02	2382	212.32	0.00	11.02	2290	215.38	0.00
100	11.34	40,730	1280.78	0.00	11.34	28,979	724.07	0.00
120	12.22	3037	527.25	0.00	12.22	2646	497.25	0.00
140	13.23	39,841	3600.33	0.09	13.23	38,859	3600.52	0.20
150	13.69	30,137	3600.36	0.26	13.69	38,969	3600.39	0.13

and

Table 17. Comparison between MST and dominating tree flow formulations using $\alpha =0.95$, $r=0.2$, $MIPGap=1\%$, and number of users $5N$ using valid inequalities.

N	MST Obj	MST B&B	CPU (s)	Gap (%)	DT Obj	B&B	CPU (s)	Gap (%)
70	4.43	155,726	3600.26	3.16	4.91	18,323	384.88	0.96
75	4.09	40,343	3600.47	14.17	4.25	43,060	1885.92	0.97
80	4.44	39,398	3600.55	17.57	4.61	39,744	3600.46	9.80

report the computational performance of the MST and DT flow formulations, plus the valid inequalities for different values of $\alpha$, with a fixed radius $r=0.2$. In general, the incorporation of valid inequalities improves tractability and helps maintain very small optimality gaps, especially for moderate values of $\alpha$. For $\alpha =0.25$ and $\alpha =0.5$ (Table 14 and Table 15), many instances are solved almost at the root node, indicating that the relaxation becomes significantly tighter once the additional inequalities are included. Also, notice that the CPU times remain moderate and grow smoothly with instance size.

For $\alpha =0.75$, the search becomes noticeably harder: both models require larger branch-and-bound trees and longer solution times. However, most of the instances are still solved optimally or near-optimally with reasonable computational effort, which confirms that the valid inequalities are useful in more challenging regimes.

For $\alpha =0.95$ (Table 17), the difficulty increases considerably. Several instances reach the time limit and the resulting optimality gaps are no longer negligible. In this regime, the DT formulation appears slightly more stable than the MST formulation, although both models become significantly harder to solve.

This is expected, since a larger $\alpha$ shifts the objective toward infrastructure cost and leaves less room for the assignment term to guide the selection of active nodes.

Finally, it is observed that these results confirm that the valid inequalities improve the single-flow formulations. They reduce the size of the search tree, improve lower bounds, and extend the range of instances that can be solved efficiently. At the same time, the experiments show that the computational difficulty still strongly depends on $\alpha$, playing a central role in shaping the complexity of the problem.

5.3. Local Branching Matheuristic

The single-flow formulation is the most reliable exact model in the experiments, but the hardest instances are still difficult to certify as optimal. Figure 3 reports this behavior by showing the evolution of the upper and lower bounds over time for representative MST and DT flow instances. In both cases, the MIP gap closes rapidly in the early phase of the search, suggesting that the model structure is well captured by the formulation. However, certifying optimality becomes increasingly difficult in more challenging instances. This motivates the use of a matheuristic designed specifically for the largest and most demanding cases.

Remark 3.

The activation vector x acts as the main structural decision layer of the problem, while the remaining variables are adjusted once x is fixed. This makes x a natural candidate for defining local branching neighborhoods.

Table 18. Performance of the local branching matheuristic.

Model	N	r	$\mathit{\alpha }$	Init.Obj.	Best.Obj.	Imp. (%)	${\mathit{t}}^{\mathit{init}}$	$\mathit{\tau }$	CPU (s)	Iter.	Impv.	${\mathit{\varphi }}_{\mathit{f}}$
MSTF	70	0.20	0.95	4.6617	4.4427	4.70	20	30	264.66	8	4	13
DTF	70	0.20	0.95	5.0248	4.9084	2.32	20	30	264.69	8	3	15
MSTF	75	0.20	0.95	4.1853	4.0952	2.15	20	30	265.30	8	6	9
DTF	75	0.20	0.95	4.3092	4.2461	1.46	20	30	261.97	8	3	15
MSTF	80	0.20	0.95	4.5434	4.4412	2.25	20	30	265.87	8	2	17
DTF	80	0.20	0.95	4.6810	4.6006	1.72	20	30	265.58	8	4	13
MSTF	100	0.20	0.95	6.2929	4.9845	20.79	20	30	268.02	8	6	9
DTF	100	0.20	0.95	5.9806	5.1709	13.54	20	30	268.09	8	7	7
MSTF	120	0.20	0.95	6.1822	5.4614	11.66	20	30	271.23	8	6	9
DTF	120	0.20	0.95	6.7784	5.4496	19.60	20	30	270.94	8	8	5
MSTF	150	0.20	0.95	8.3723	6.0012	28.32	20	30	247.42	8	8	5
DTF	150	0.20	0.95	8.5108	6.1406	27.85	20	30	276.32	8	8	5

reports the computational performance of the proposed local branching matheuristic for the hardest instances, with up to $\left|V\right|=150$ nodes, fixed $r=0.20$ and $\alpha =0.95$. The table reports the quality of the initial solution, the best solution found, the relative improvement, the time used to generate the initial solution, the time limit per local branching subproblem, the total CPU time, the number of iterations, the number of improving iterations, and the final neighborhood size.

From a scalable point of view, the results report that the local branching procedure keeps the computational cost relatively stable as the instance size increases. Notice that each run uses a fixed number of iterations and a fixed time limit per neighborhood subproblem. The latter makes the total computational cost predictable. Moreover, the procedure continues to produce improvements for the largest instances. For $N=150$, the improvement reaches $28.32\%$ for MST_F and $27.85\%$ for DT_F. This indicates that the restricted neighborhoods remain effective even when the full exact model becomes harder. The role of local branching is therefore complementary: it is used when good feasible solutions are more useful than spending the full time on closing the final gap.

Notice that the results also show that the matheuristic significantly improves the initial solutions obtained for all tested instances. In particular, the largest ones. The relative improvement is often notable, exceeding $25\%$. The total CPU time remains stable through the tested instance sizes, and the number of successful iterations is relatively high, indicating that the local branching strategy explores the solution space effectively. This is especially relevant in the most difficult operation schemes, where the exact solver alone may fail to produce high-quality solutions within the time limit. Furthermore, the comparison between MST_F and DT_F reveals an important difference. Although both models benefit from mathematical simplicity, DT_F remains harder because domination constraints restrict the feasible region more. As expected from the theoretical dominance relation established in Section 4, the best objective values obtained for DT_F are systematically higher than those for MST_F. In addition, the percentage improvements achieved for DT_F are often slightly smaller or more difficult to obtain, reflecting the greater combinatorial burden imposed by domination. In general, Table 18 shows that the proposed local branching framework is a robust and scalable complement to the exact method, particularly for large and difficult instances where direct branch-and-bound becomes impractical.

5.4. Structural Analysis of Flow Solutions

The structural plots are interpreted through five quantities: the number of active nodes, the total tree length, the tree diameter, the average degree, and the average user distance. The structural properties of the solutions produced by MST_F and DT_F are considered next. The goal of this analysis is to better understand how the trade-off between backbone-construction cost and user-assignment cost influences the topology of the resulting network structures. Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 report the behavior of several structural metrics as the number of nodes N increases. In all cases, both formulations are evaluated in the same set of instances.

The metrics considered are as follows. The size of the backbone, defined as $\left|\right\{i\in V:x_i=1\left\}\right|$, measures the number of active nodes and, therefore, the size of the deployed infrastructure. The total length of the tree, given by ${\sum }_{(i,j)\in A}{c}_{ij}{z}_{ij}$, measures the total cost of building the structure. The tree diameter is the maximum shortest-path distance between any pair of selected nodes in the induced tree and reflects how spread out the backbone is. The average degree measures the mean number of incident selected edges per active node and helps distinguish between chain-like and hub-like structures. Finally, the average user distance, computed as ${\displaystyle \frac{1}{\left|K\right|}}{\sum }_{k\in K}{\sum }_{i\in V}{d}_{ki}{w}_{ki}$, captures the average assignment cost and, therefore, the quality of user service. Smaller values suggest better proximity between users and active nodes.

Figure 4 illustrates the comparison between MST_F and DT_F for different values of $\alpha$. It can be observed that for small values of $\alpha$, both formulations emphasize user service and therefore produce relatively large and spatially distributed backbones. In this network regime, the two solutions are visually similar. As $\alpha$ increases, the cost of the backbone becomes significantly more important, and the structural gap between the two models becomes more evident. MST_F generates smaller and more compact backbones, whilst DT_F remains more spatially distributed because of the domination constraints. The figure already shows the main pattern that will appear in the larger experiments: small $\alpha$ values produce similar trees. Larger values separate the MST_F and DT_F designs.

Figure 5 reports the structural metrics for $\alpha =0.25$. In this coverage regime, MST_F and DT_F exhibit nearly identical behaviors across all metrics. The backbone sizes increase approximately linearly with N. The average user distances decrease as the network becomes denser, and both the length and the diameter of the trees increase similarly.

The average degree is nearly two, which is compatible with the tree-like nature of the output solutions. The close overlap of the curves suggests that for small values of $\alpha$, the domination constraints become redundant, and both models produce almost the same network structures. Figure 6 presents the structural behavior for $\alpha =0.5$, where the backbone-construction cost and the user-assignment cost seem to be more balanced. Under this network operating regime, a mild divergence appears from the solutions of DT_F as they start activating slightly more nodes and therefore produce somewhat larger backbones and longer trees than MST_F. The diameter of the tree is slightly larger under DT_F. This also reflects a more distributed structure. Although the differences remain moderate, the two models still present the same qualitative behavior. This regime can be interpreted as an intermediate zone in which domination starts to affect but has not yet become the main structural characteristic of the obtained solution.

For $\alpha =0.75$, Figure 7 reports a significantly clearer divergence between the two formulations. MST_F produces more compact solutions with smaller backbones and lower total tree lengths, whereas DT_F requires additional active nodes to satisfy the domination constraints. As a consequence, the DT backbones become larger and more dispersed, and their diameters are greater. At the same time, DT_F achieves slightly lower average user distance, reflecting better spatial service at the expense of more expensive infrastructure. This figure presents evidence of a regime in which domination has a clear structural effect and changes the backbone design.

Figure 8 corresponds to the extreme regime where the objective is dominated by backbone-construction costs. The structural differences between MST_F and DT_F are more systematic and pronounced. MST_F activates the minimum number of nodes needed to form a low-cost connected structure, whereas DT_F remains substantially larger because it must also dominate the network. The latter produces clear differences in the size of the backbone, the total length of the tree, and the diameter of the tree. Simultaneously, DT_F continues to provide better user proximity, as this is reflected by its smaller average user distance.

In general, the figure confirms that domination is far from redundant in this regime and leads to fundamentally different network designs.

Taken together, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 reveal a consistent structural pattern. For small values of $\alpha$, MST and DT are almost indistinguishable both visually and quantitatively. As $\alpha$ increases, the two formulations separate: MST favors compactness and low infrastructure cost, while DT preserves broader spatial coverage through a larger active backbone. This supports the idea that the practical relevance of domination is strongly dependent on the operating regime.

5.5. Joint Sensitivity with Respect to r and $\alpha$

The joint effect of the radius r and the weighting parameter $\alpha$ is analyzed on the behavior of the flow models. This part of the numerical experiment aims to find the regions of the $(\alpha ,r)$ space where MST_F and DT_F coincide, are similar, or clearly diverge. For each pair $(\alpha ,r)$, the objective function value, the CPU time in seconds, and the difference between the objective function values of the two formulations are evaluated. In this subsection, $\alpha$ is varied for the full interval $[0, 1]$. This complements the representative values used in the previous tables and allows us to examine the MST-DT relationship across the complete weighting range. Figure 9 presents a three-dimensional sensitivity analysis on $(\alpha ,r)$. The left panel reports the objective surfaces of MST_F and DT_F. The right panel reports the corresponding CPU time surfaces. Notice that a clear pattern emerges. For small values of $\alpha$ and moderate-to-large values of r, the objective surfaces of the two models overlap or remain very close. This indicates that in dense graphs and coverage-oriented operating network regimes, the domination constraints become much less restrictive. In contrast, as $\alpha$ increases and/or r decreases, the gap between the surfaces tends to be more pronounced, and DT_F consistently yields higher objective values.

The CPU time surfaces show a related but distinct effect. Both formulations remain computationally tractable in denser regimes, but DT_F tends to require more computational effort in sparse graphs and for larger values of $\alpha$. This reflects the combined effect of a more constrained feasible region and a more difficult combinatorial structure. Rather than interpreting this pattern as a formal phase transition, this is viewed as strong empirical evidence of a transition region separating operating regimes in which domination is practically negligible from regimes in which it has a substantial structural and computational impact.

Figure 10 shows the region in which both formulations produce identical objective function values. This region is concentrated mainly in the lower-$\alpha$ and larger-r parameter space. The main idea is that domination may become effectively redundant in dense and coverage-oriented settings. Outside of this region, the difference between MST_F and DT_F increases steadily. Finally, Figure 11 shows the empirical boundary that separates the regions of coincidence from the regions of divergence in the $(\alpha ,r)$ space. For small $\alpha$ and sufficiently large r, both models coincide. Beyond this frontier, domination induces a systematic gap between the two formulations. This boundary should be interpreted as empirical rather than theoretical; however, it provides a useful operational summary of when domination matters and when it does not.

Notice that the boundary shown in Figure 11 should not be regarded as a proven threshold but as an empirical summary of the tested instances, indicating where the domination constraints appear to be active and where they appear to be redundant. In general, computational experiments support four main conclusions. First, the single-flow formulation is the most effective exact model among those considered. Second, the proposed valid inequalities substantially strengthen this formulation, especially in moderate regimes. Third, the local branching methodology provides a practical mechanism for handling the hardest large-scale instances. Fourth, the structural similarity between MST and DT depends strongly on the pair $(\alpha ,r)$: in dense and coverage-oriented regimes, the two models are often nearly indistinguishable, whereas in sparse and backbone-cost-dominated regimes, they lead to clearly different network designs.

Figure 9, Figure 10 and Figure 11 should be read as complementary views of the same structural phenomenon. Figure 9 shows how the objective and CPU time vary jointly with $\alpha$ and r. Whilst Figure 10 highlights the region where the MST and DT objective values coincide, Figure 11 summarizes the empirical boundary between coincidence and divergence. These figures indicate that domination tends to be less restrictive for larger radii and smaller values of $\alpha$, whilst it becomes more relevant in sparse and infrastructure-oriented network regimes.

It is important to emphasize that the computational study is designed as a controlled formulation-level and structural analysis rather than as a complete benchmarking exercise against all existing heuristics and metaheuristics for related backbone problems. Consequently, synthetic Euclidean instances are used because they allow the number of nodes, radius, and weighting parameter to be varied systematically, preserving a fair comparison between MST- and DT-based models under identical data. This controlled setting is particularly useful for identifying the regime zones in which domination constraints are structurally active or redundant. To support this analysis, explicit similarity metrics are computed between MST_F and DT_F solutions. Let $X^{\mathrm{MST}}=\{i\in V:x_i^{\mathrm{MST}}=1\}$ and $X^{\mathrm{DT}}=\{i\in V:x_i^{\mathrm{DT}}=1\}$ denote the active-node sets selected by the two models. The active-node Jaccard similarity metric is defined as

$$J_X={\displaystyle \frac{|X^{\mathrm{MST}}\cap X^{\mathrm{DT}}|}{|X^{\mathrm{MST}}\cup X^{\mathrm{DT}}|}}.$$

The metric satisfies $0\le J_X\le 1$, where $J_X=1$ means that both models select exactly the same active nodes, whilst the smaller values indicate lower structural overlaps. In addition, the relative objective function difference is computed by

$${\mathrm{\Delta }}_{\mathrm{obj}}={\displaystyle \frac{|z_{\mathrm{DT}}^{★}-z_{\mathrm{MST}}^{★}|}{max\{1,|z_{\mathrm{MST}}^{★}\left|\right\}}}.$$

Thus, $J_X$ measures structural similarity with respect to the selected backbone nodes. Similarly, ${\mathrm{\Delta }}_{\mathrm{obj}}$ measures how close the two optimal objective function values are.

The similarity metrics indicate that in dense and service-oriented regions, $J_X$ is high and ${\mathrm{\Delta }}_{\mathrm{obj}}$ is close to zero. This suggests that the two formulations select similar active backbones and obtain nearly identical objective function values. In contrast, in sparse and infrastructure-oriented network regimes, $J_X$ decreases and ${\mathrm{\Delta }}_{\mathrm{obj}}$ increases, clearly showing that the domination constraints significantly modify the selected backbone. Therefore, the observed transition is supported by visual inspection of the solution surfaces.

To complement the visual analysis of the $(\alpha ,r)$ space,

Table 19. Similarity metrics between MST_F and DT_F solutions for representative values of $\alpha$ and r. Instance of size $N=50$.

$\mathit{\alpha }$	r	${\mathit{J}}_{\mathit{X}}$	${\mathit{\Delta }}_{\mathbf{obj}}$	Interpretation
0.25	0.20	1.000	0.000	High similarity
0.50	0.20	1.000	0.000	High similarity
0.75	0.20	1.000	0.000	High similarity
0.95	0.20	0.391	0.116	Clear divergence
0.25	0.30	0.980	0.000	High similarity
0.50	0.30	1.000	0.000	High similarity
0.75	0.30	1.000	0.000	High similarity
0.95	0.30	0.368	0.059	Clear divergence

presents the proposed similarity metrics obtained for representative parameter values. The experiment uses an instance of size $N=50$ and compares MST_F and DT_F for two communication radii, $r=0.20$ and $r=0.30$, and four values of the weighting parameter $\alpha$. The metric $J_X$ measures the overlap between the active-node sets selected by both formulations, while ${\mathrm{\Delta }}_{\mathrm{obj}}$ measures the relative difference between their optimal objective values.

The results show that for $\alpha \in \{0.25,0.50,0.75\}$, both formulations are almost indistinguishable in the tested instances. In these cases, $J_X$ is equal to or very close to one, and ${\mathrm{\Delta }}_{\mathrm{obj}}$ is zero, indicating that MST_F and DT_F select nearly the same active nodes and achieve the same objective value. This provides quantitative support for the collapse behavior observed in the solution surfaces and structural plots.

The only clear divergence appears for $\alpha =0.95$, where the objective becomes strongly infrastructure-oriented. In this regime, $J_X$ decreases to $0.391$ for $r=0.20$ and $0.368$ for $r=0.30$, while ${\mathrm{\Delta }}_{\mathrm{obj}}$ increases to $0.116$ and $0.059$, respectively. This confirms that domination constraints become structurally relevant mainly when the model prioritizes backbone-construction cost over user proximity. Therefore, the structural transition is supported not only by visual inspection but also by explicit similarity indicators.

5.6. Sensitivity with Respect to the Number of Users

To evaluate the effect of the user layer, an additional sensitivity analysis is performed by varying the number of users as $\left|K\right|\in \{N,3N,5N,10N\}$. In these experiments, the number of candidate backbone nodes is fixed at $N=50$, while the communication radius r and the weighting parameter $\alpha$ are varied across representative values. This design allows us to assess whether the baseline choice $\left|K\right|=5N$ affects the structural relation between MST_F and DT_F solutions. The users are interpreted as external demand points, traffic sources, or clients that must be assigned to the selected backbone. Therefore, increasing $\left|K\right|$ increases the density of the demand layer and the aggregate contribution of the user-assignment term in the objective function. This experiment separates the effect of user density from the effects of r and $\alpha$, and shows how the demand layer changes the objective value, the number of active nodes, the average assignment distance, and $J_X$.

Table 20. Sensitivity with respect to the number of users $(N=50;\alpha =0.25,radius=0.2)$.

$\left|\mathit{K}\right|$	Model	Obj.	Backbone Size	Avg. User Dist.	${\mathit{J}}_{\mathit{X}}$	CPU (s)
N	MSTF	3.672	30	0.0692	1.000	1.65
N	DTF	3.672	30	0.0692	1.000	1.52
$3N$	MSTF	9.009	45	0.0691	1.000	0.72
$3N$	DTF	9.009	45	0.0691	1.000	0.62
$5N$	MSTF	14.379	49	0.0699	1.000	0.73
$5N$	DTF	14.379	49	0.0699	1.000	0.61
$10N$	MSTF	28.580	50	0.0727	1.000	0.94
$10N$	DTF	28.580	50	0.0727	1.000	0.61

,

Table 21. Sensitivity with respect to the number of users $(N=50;\alpha =0.5,radius=0.2)$.

$\left|\mathit{K}\right|$	Model	Obj.	Backbone Size	Avg. User Dist.	${\mathit{J}}_{\mathit{X}}$	CPU (s)
N	MSTF	3.632	22	0.0828	0.840	8.29
N	DTF	3.654	24	0.0792	0.840	16.92
$3N$	MSTF	7.619	38	0.0711	1.000	3.48
$3N$	DTF	7.619	38	0.0711	1.000	1.69
$5N$	MSTF	11.250	44	0.0703	1.000	2.17
$5N$	DTF	11.250	44	0.0703	1.000	1.19
$10N$	MSTF	20.805	50	0.0727	1.000	1.91
$10N$	DTF	20.805	50	0.0727	1.000	1.15

,

Table 22. Sensitivity with respect to the number of users $(N=50;\alpha =0.75,radius=0.2)$.

$\left|\mathit{K}\right|$	Model	Obj.	Backbone Size	Avg. User Dist.	${\mathit{J}}_{\mathit{X}}$	CPU (s)
N	MSTF	3.044	16	0.1072	0.636	94.29
N	DTF	3.321	20	0.1026	0.636	10.90
$3N$	MSTF	5.594	29	0.0841	1.000	102.27
$3N$	DTF	5.594	29	0.0841	1.000	14.48
$5N$	MSTF	7.823	35	0.0824	1.000	48.35
$5N$	DTF	7.823	35	0.0824	1.000	32.10
$10N$	MSTF	12.954	46	0.0739	1.000	24.93
$10N$	DTF	12.954	46	0.0739	1.000	7.16

,

Table 23. Sensitivity with respect to the number of users $(N=50;\alpha =0.95,radius=0.2)$.

$\left|\mathit{K}\right|$	Model	Obj.	Backbone Size	Avg. User Dist.	${\mathit{J}}_{\mathit{X}}$	CPU (s)
N	MSTF	0.988	1	0.3951	0.000	4.69
N	DTF	2.608	18	0.1305	0.000	2.93
$3N$	MSTF	2.473	8	0.2117	0.350	176.30
$3N$	DTF	3.205	19	0.1232	0.350	17.17
$5N$	MSTF	3.426	11	0.1709	0.391	387.29
$5N$	DTF	3.825	21	0.1140	0.391	26.17
$10N$	MSTF	5.213	19	0.1217	0.692	1552.80
$10N$	DTF	5.292	25	0.1017	0.692	120.93

,

Table 24. Sensitivity with respect to the number of users $(N=50;\alpha =0.25,radius=0.3)$.

$\left|\mathit{K}\right|$	Model	Obj.	Backbone Size	Avg. User Dist.	${\mathit{J}}_{\mathit{X}}$	CPU (s)
N	MSTF	3.528	28	0.0691	1.000	3.45
N	DTF	3.528	28	0.0691	1.000	2.93
$3N$	MSTF	8.915	45	0.0691	1.000	1.54
$3N$	DTF	8.916	45	0.0691	1.000	1.18
$5N$	MSTF	14.286	48	0.0699	0.980	1.42
$5N$	DTF	14.285	49	0.0699	0.980	1.31
$10N$	MSTF	28.486	50	0.0727	1.000	1.58
$10N$	DTF	28.488	50	0.0727	1.000	1.35

,

Table 25. Sensitivity with respect to the number of users $(N=50;\alpha =0.5,radius=0.3)$.

$\left|\mathit{K}\right|$	Model	Obj.	Backbone Size	Avg. User Dist.	${\mathit{J}}_{\mathit{X}}$	CPU (s)
N	MSTF	3.495	22	0.0771	1.000	16.18
N	DTF	3.495	22	0.0771	1.000	19.92
$3N$	MSTF	7.431	38	0.0711	1.000	6.56
$3N$	DTF	7.431	38	0.0711	1.000	3.94
$5N$	MSTF	11.062	44	0.0703	1.000	4.44
$5N$	DTF	11.062	44	0.0703	1.000	3.65
$10N$	MSTF	20.617	50	0.0727	1.000	3.30
$10N$	DTF	20.617	50	0.0727	1.000	3.40

,

Table 26. Sensitivity with respect to the number of users $(N=50;\alpha =0.75,radius=0.3)$.

$\left|\mathit{K}\right|$	Model	Obj.	Backbone Size	Avg. User Dist.	${\mathit{J}}_{\mathit{X}}$	CPU (s)
N	MSTF	2.940	14	0.1116	0.933	547.00
N	DTF	2.944	15	0.1044	0.933	190.01
$3N$	MSTF	5.504	26	0.0830	1.000	85.04
$3N$	DTF	5.504	26	0.0830	1.000	100.36
$5N$	MSTF	7.629	32	0.0770	1.000	197.12
$5N$	DTF	7.629	32	0.0770	1.000	47.14
$10N$	MSTF	12.672	46	0.0739	1.000	31.89
$10N$	DTF	12.672	46	0.0739	1.000	16.74

and

Table 27. Sensitivity with respect to the number of users $(N=50;\alpha =0.95,radius=0.3)$.

$\left|\mathit{K}\right|$	Model	Obj.	Backbone Size	Avg. User Dist.	${\mathit{J}}_{\mathit{X}}$	CPU (s)
N	MSTF	0.988	1	0.3951	0.000	7.17
N	DTF	1.932	8	0.1727	0.000	53.94
$3N$	MSTF	2.377	8	0.1985	0.214	1241.83
$3N$	DTF	2.738	9	0.1601	0.214	849.83
$5N$	MSTF	3.334	11	0.1636	0.389	3600.30
$5N$	DTF	3.529	14	0.1267	0.389	2142.90
$10N$	MSTF	5.123	18	0.1239	0.947	3600.51
$10N$	DTF	5.155	19	0.1159	0.947	3600.36

show that the number of users has a visible effect on the magnitude of the objective value and on the average user distance, as expected, because both quantities depend directly on the assignment layer. For small and moderate values of $\alpha$, the MST_F and DT_F solutions remain almost identical across most values of $\left|K\right|$. In particular, for $\alpha =0.25$ and $\alpha =0.50$, the Jaccard similarity $J_X$ is equal to or very close to one in most cases, indicating that both formulations select essentially the same active backbone nodes. Using larger values of $\alpha$, say $\alpha =0.95$, and smaller values of $\left|K\right|$ leads to clear structural differences between MST_F and DT_F. The low values of $J_X$ and significantly different backbone sizes confirm that domination constraints become relevant in infrastructure-oriented network regimes. However, as $\left|K\right|$ increases, the assignment component of the objective becomes more influential in aggregate, and the MST_F trends are activating additional nodes to reduce user-assignment costs. As a result, the similarity between MST_F and DT_F may increase for larger user sets, especially when $\left|K\right|=10N$.

Notice that these results make $\left|K\right|=5N$ a reasonable baseline for the main experiments. This value creates a demand that is dense enough to make the assignment term meaningful while still preserving the structural differences between MST_F and DT_F in infrastructure-oriented network regimes. The sensitivity analysis also refines the interpretation of the previous results: the MST_F-DT_F transition is mainly dominated by $\alpha$ and r, but it is also influenced by the number of users. In particular, a larger number of users can make MST_F behave more similarly to DT_F because the assignment-cost component encourages the activation of more spatially distributed backbone nodes.

6. Discussion and Conclusions

The main conclusion of this study is that the role of domination in the design of the backbone network is not fixed but strongly depends on the network’s operating regime. In sparse environments, domination plays a significant structural role because it forces the backbone to remain close to the rest of the network, typically requiring the activation of additional nodes and, consequently, increasing infrastructure costs. In contrast, in denser networks, this requirement may become much less restrictive, and the backbones produced by the DT model tend to coincide with or remain very close to those obtained by the classical spanning tree model (MST). The additional sensitivity analysis with respect to the number of users further indicates that this transition is also modulated by the density of the demand layer, since larger user sets increase the aggregate importance of the assignment-cost component.

From a computational perspective, the experiments show that single-flow formulations provide the best balance between formulation strength and tractability. The latter advantage becomes even more useful when the proposed valid inequalities are included. For the hardest instances, the local branching matheuristic also proves effective, delivering high-quality solutions in network regimes where direct exact methods become computationally prohibitive. Putting it all together, these results provide both a practical modeling framework and a clearer structural understanding of when domination constraints meaningfully affect backbone design in geometric communication networks.

Quantitatively, the computational results show that the single-flow formulation is the most effective exact model among the tested alternatives. In the main benchmark experiments, it solved instances with up to $N=100$ nodes with a zero optimality gap in most regimes, while the strengthened flow formulation allowed the analysis to be extended up to $N=150$ nodes. In the most difficult regime, corresponding to $\alpha =0.95$ and $r=0.20$, the local branching matheuristic produced substantial improvements over the initial feasible solutions. For the largest tested instances with $N=150$, the improvement reached $28.32\%$ for MST_F and $27.85\%$ for DT_F. The new similarity and user-sensitivity metrics also confirm that MST_F and DT_F tend to coincide in service-oriented regimes, whereas their divergence becomes clearer when the objective is dominated by infrastructure cost. These numerical results support the practical value of the proposed formulation and matheuristic components.

Several directions for future research naturally arise. These include extensions to dynamic or multi-period settings, the incorporation of uncertainty in connectivity or user-demand patterns, and the study of richer backbone structures under robustness, reliability, or survivability requirements. Another important direction is to validate the proposed framework using real-world or carefully calibrated realistic communication network data. In addition, future work should compare the proposed exact and matheuristic approaches with specialized heuristics and metaheuristics from the literature under a common benchmarking protocol. Such extensions would further assess the practical applicability and competitiveness of the framework beyond the controlled synthetic Euclidean setting considered in this study.