The Budgeted Labeled Minimum Spanning Tree Problem

Abstract

In order to reduce complexity when designing multi-media communication networks, researchers often consider spanning tree problems defined on edge-labeled graphs. The earliest setting addressed in the literature aims to minimize the number of different media types, i.e., distinct labels, used in the network. Despite being extensively addressed, such a setting completely ignores edge costs. This led to the definition of more realistic versions, where budgets for the total cost, or the number of distinct labels allowed, were introduced. In this paper, we introduce and prove the NP-hardness of the Budgeted Labeled Minimum Spanning Tree problem, consisting in minimizing the cost of a spanning tree while satisfying specified budget constraints for each label type. This problem combines the challenges of cost efficiency and label diversity within a fixed budgetary framework, providing a more realistic and practical approach to network design. We provide three distinct mathematical programming formulations of the problem and design a Lagrangian approach to derive tighter lower bounds for the optimal solution of the problem. The performances of the proposed methods are assessed by conducting a series of computational experiments on a variety of randomly generated instances, which showed how the complexity of the problem increases as the size of the network, as well as the number of labels, increase and the budget restrictions are tightened.

1. Introduction

Designing effective multi-media communication networks is becoming increasingly relevant in today’s digital era. One of the most studied network design problems is the well-known Minimum Spanning Tree (MST) problem, in which a cost is associated with each edge of the network and the objective is to identify a spanning tree, namely a connected acyclic subgraph spanning all the nodes of the network, of minimum cost. The applications of this problem extend beyond the network design scenarios, encompassing several fields such as constructing highways or railroads connecting multiple cities; laying pipelines to connect offshore drilling sites, refineries, and consumer markets; designing local access networks; and even making electric wire connections on a control panel [1]. Researchers have delved into a wide range of spanning tree problems, particularly those associated with edge-labeled graphs [2].

One of the first proposed natural extensions of the MST is the Minimum Label Spanning Tree (MLST) problem, consisting in finding a spanning tree in a connected, undirected graph with labeled (or colored) edges, while minimizing the number of distinct labels used. Such an extension models more realistic scenarios in which different communication media, including fiber optics, cable, microwave, or telephone lines, may be implemented. In such contexts, building a spanning tree that relies on a smaller number of media types is more desirable, as it is associated with a reduced complexity of the communication process. However, the MLST does not take into account the cost of the installed edges. For this reason, more realistic variants, incorporating spanning tree cost restrictions, as well as constraints on the number of distinct labels permitted, have been proposed [3]. In the former case, one can have a maximum cost budget available to establish the links of the network and focus on using as few different media types as possible, represented by distinct labels, aiming at reducing communication complexity. In the latter case, instead, the number of different media types is upper bounded by a provided budget and the construction costs are minimized.

While this line of research has been extensively pursued, it often overlooks a critical aspect: in numerous applications, the availability of a specific type of connection may be limited and the outcome may consequently depend on the number of available connections supporting each media type.

Furthermore, as considered in previous works [4], multiple labels can be associated with each edge, representing the requirement of supporting different technologies, as well as different transmission frequencies, when establishing a connection between two endpoints. In this case, a label set is associated with each edge of the network and selecting such an edge in the solution entails the implementation of all the technologies supported by the related connection, as indicated by its label set. Budget constraints also enable the decision maker to express preferences on the available media types, thus influencing the homogeneity of the network.

To model this scenario from a cost-minimization perspective, this paper introduces the Budgeted Labeled Minimum Spanning Tree (BLMST) problem, which extends the MST problem by considering budget constraints for each label. The BLMST problem aims to minimize the total cost of the spanning tree while ensuring that, for each type of label, the number of edges using that label does not exceed the specified budget. Unlike the MST problem, which can be efficiently solved in polynomial time, the BLMST problem is NP-hard due to budget constraints. We provide three distinct mathematical programming formulations for the BLMST problem and propose a Lagrangian approach to derive tight lower bounds on the optimal solution value. Through comprehensive computational experiments conducted on randomly generated instances, we provide a comparative analysis of the formulations and detect relevant trends in terms of solution costs and computational times, as a function of several characteristics of the instances.

The rest of the paper is divided into the following sections: review of the literature on MST-related optimization problems, with a focus on those defined on edge-colored graphs and the ones equipped with budget constraints (Section 2); formal definitions and notations used in our study to describe the problem (Section 3); proof of the NP-hardness of the BLMST problem and presentation of the mathematical models proposed to solve it (Section 4 and Section 5); properties of the BLMST problem (Section 6); description of a Lagrangian approach designed to derive tighter lower bounds, eventually with upper bounds, for the problem (Section 7); computational experiments conducted to validate the proposed approaches (Section 8); and summary of the key findings, contributions, and implications of this study (Section 9).

2. Literature Review

Numerous problems defined on edge-colored graphs have been addressed in the literature, including extensions of the traveling salesman problem [5] and the Hamiltonian cycle problem [6], as well as the labeled perfect matching problem [7].The aforementioned MLST problem was first introduced and proven to be NP-complete in [8], along with an exact algorithm based on the $A^{*}$ paradigm and two heuristic algorithms. Since then, several contributions have been provided, including exact approaches [9], heuristics [10], and metaheuristics [11,12]. In [13], Consoli and Moreno Pérez introduced a hybrid local search algorithm for the MLST, combining variable neighborhood search and simulated annealing paradigms, while an intelligent optimization approach was proposed in [14], integrating variable neighborhood search with machine learning, statistics, and soft computing techniques.

Generalizations of the problem have subsequently been investigated. In this context, Chen et al. introduced the Generalized Minimum Labeling Spanning Tree (GMLST) problem [15], for which Silva et al. recently provided a compact binary integer programming formulation and an analysis of the associated polytope [16]. The GMLST has the same objective as the MLST, i.e., minimizing the number of used labels, but it is defined on graphs whose edges are associated with multiple labels. Chwatal and Raidl adapted single commodity flow and Miller–Tucker–Zemlin-based formulations to the GMLST problem and proposed new formulations, together with strengthening cuts [17]. Since a single label can be selected when including an edge in the solution, the GMLST problem can be reduced to the MLST problem by replacing each edge $e\in E$ with a set of parallel edges whose cardinality corresponds to the number of labels associated with e [16]. Afterwards, Cerrone et al. introduced the Strong Generalized Minimum Label Spanning Tree (SGMLST) problem [4]. This further generalization of both the MLST and GMLST problems aims at minimizing the number of labels used while designing a spanning tree in contexts where selecting an edge entails the usage of all the labels associated with it. The authors proposed a mathematical programming formulation and three heuristic approaches for the problem, based on greedy, carousel greedy, and pilot-method [18] techniques.

In [19], a more realistic extension of the MLST problem, namely the Label-Constrained Minimum Spanning Tree (LCMST) problem, has been introduced. The LCMST consists in finding the min-cost spanning tree of a graph that does not exceed a specified number of labels. The authors proposed a mixed integer programming formulation of the problem, a local search heuristic, and a genetic algorithm to solve it. In the same paper, a further extension of the MLST, named Cost Constrained Minimum Spanning Tree (CCMST) problem, has been introduced. Specularly to the LCMST, the CCMST aims to find a spanning tree using the minimum number of labels without exceeding a given cost budget. Both problems have been shown to be NP-complete. In [3], a variable neighborhood search method has been proposed for the CCMST and then adapted for the LCMST as well.

Other extensions of the MLST problem include the Minimum Labeling Steiner Tree (MLST) problem [20], aiming to minimize the number of different labels while covering specified vertices of the graph, as well as the Minimum Spanning Tree with Conflicts (MSTC), involving conflicting pairs of edges, among which at most one of each pair can belong to a feasible solution. This problem was initially proposed as “MST with disjunctive constraints” in [21,22], where conflicts between edges are represented in a conflict graph. Since then, several solution approaches have been proposed in the literature [23,24,25,26,27].

3. Definitions and Notations

The BLMST problem, addressed in this paper, is defined on an undirected and weighted graph $G=(V,E)$, together with a set L of distinct labels. When referring to an edge of G, we adopt two alternative notations: when we do not need to refer to edge endpoints, we write $e\in E$; otherwise, we use the notation $\{i,j\}\in E$. Additionally, let us define the neighborhood $\delta \left(i\right)$ of any node $i\in V$ as the set containing all the nodes that are directly connected to node i in G, formally $\delta \left(i\right)=\{j\in V:\{i,j\}\in E\}$.

Each edge $e\in E$ is associated with a cost $c_e$, contributing to the total cost of the spanning tree if e is selected, as well as with a subset of labels, specified by an edge-labeling function $\varphi :E\to \mathcal{P}\left(L\right)$, where $\mathcal{P}\left(L\right)$ indicates the set of all possible subsets of L. Note that the same label can be associated with multiple edges and vice versa. Let us denote by $E_{\ell }$ the set of edges labeled with ℓ, namely $E_{\ell }=\{e\in E:\ell \in \varphi \left(e\right)\}$. For each label $\ell \in L$, the number of edges in the spanning tree with label ℓ must not exceed a given budget $B_{\ell }$. The objective of the BLMST problem is to find a minimum spanning tree that minimizes the total coverage cost while satisfying the budget constraints for each label. Formally, the BLMST problem asks for a spanning tree $T=(V,E_T)$ of G such that:

• ${\sum }_{e\in E_T}{c}_e\le {\sum }_{e\in E_{\tilde{T}}}{c}_e\forall \mathrm{spanning}\mathrm{tree}\tilde{T}\mathrm{of}G$;
• $|E_{\ell }\cap E_T|\le B_{\ell }\forall \ell \in L$, i.e., budget constraints are satisfied for each label.

4. Complexity

In this section, we prove the NP-hardness of the BLMST problem by providing a polynomial reduction from the Degree Constrained Minimum Spanning Tree (DCMST) problem, which is known to be NP-hard [28]. The DCMST problem consists in finding a min-cost spanning tree of a given weighted graph $G=(V,E)$ while ensuring that the degree of each node does not exceed a specified integer k, i.e., $\left|\delta \right(v\left)\right|\le k,\forall v\in V$.

The decision version of the DCMST and BLMST problems are given below.

${\mathbf{DCMST}}_{\mathbf{d}}$: Given an undirected and weighted graph $G=(V,E)$, a non-negative integer k, and a cost threshold C, determine whether there exists a spanning tree of G of cost at most C, where the degree of each node is at most k.

BLMST${}_{\mathbf{d}}$: Given an undirected and weighted graph $G=(V,E)$, a set of labels L, an edge-labeling function $\varphi :E\to \mathcal{P}\left(L\right)$, a budget $B_{\ell }$ for each label $\ell \in L$m. and a cost threshold C, determine if there exists a spanning tree of G of cost at most C, s.t. the number of edges in $E_{\ell }$ is at most $B_{\ell }$ for each $\ell \in L$.

Proposition 1.

Given an instance $I_D=(G,k,C)$ of DCMST${}_{\mathrm{d}}$, where $G=(V,E)$ is a weighted graph, k is an integer, and C is a cost threshold, it holds that DCMST${}_d$${\le }_{\mathbf{p}}$ BLMST${}_{\mathrm{d}}$.

Proof. 

We prove a polynomial-time reduction from any instance $I_D=(G,k,C)$ of DCMST${}_{\mathrm{d}}$ to an instance $I_B=(G^{\prime },L,\varphi ,B,C^{\prime })$ of BLMST${}_{\mathrm{d}}$. Let $G=(V,E)$ be the graph in $I_D$. In order to build $I_B$, let us define $G^{\prime }=(V^{\prime },E^{\prime })$, with $V^{\prime }=V$ and $E^{\prime }=E$. Furthermore, let L contain a distinct label for each node in G, i.e., $L=\{{\ell }_v:v\in V\}$, and assign a unique label ${\ell }_v$ to each node $v\in V$; for each edge $e=\{u,v\}\in E$, define $\varphi \left(e\right)=\{{\ell }_u,{\ell }_v\}$, meaning that each edge is associated with both the labels of its endpoints; set the budget $B_{{\ell }_v}$ of each label ${\ell }_v\in L$ to k. This models the degree constraint in DCMST${}_{\mathrm{d}}$; finally, impose the same cost threshold for the BLMST${}_{\mathrm{d}}$ instance, i.e., $C^{\prime }=C$. This reduction maps each feasible solution of DCMST${}_{\mathrm{d}}$($I_D$) to a feasible solution of BLMST${}_{\mathrm{d}}$($I_B$) with an equivalent cost and vice versa. Indeed, the degree constraint in DCMST${}_{\mathrm{d}}$ is effectively captured by the label budgets in BLMST${}_{\mathrm{d}}$. Therefore, a spanning tree of G with a total cost at most C and maximum node degree k corresponds to a spanning tree in $G^{\prime }$ with a total cost at most $C^{\prime }$ s.t. the number of selected edges in $|E_{\ell }|$ is at most k, which corresponds to $B_{\ell }$, for each label $\ell \in L$; thus, all the budget constraints are satisfied. Conversely, via construction, a spanning tree of $G^{\prime }$ with a total cost at most $C^{\prime }$ never uses more than $B_{\ell }=k$ edges labeled with ℓ, $\forall \ell \in L$. This means that the degree of each node $v\in V^{\prime }$ is always at most k; thus, this represents a feasible solution for the DCMST${}_{\mathrm{d}}$ of cost at most C. As a result, DCMST${}_{\mathrm{d}}$ polynomially reduces to BLMST${}_{\mathrm{d}}$.    □

Figure 1 visually represents the reduction process from DCMST to BLMST. Initially, in Figure 1a,b, we assume the cost of each edge to be one for simplicity. It is essential to note, however, that this reduction is generalizable to cases where edges have varying positive integer costs. Figure 1a shows an instance of the DCMST problem on a graph, where the maximum degree constraint is $k=2$. This figure establishes the baseline scenario for the reduction, depicting a graph where each edge is assigned a cost of one. In Figure 1b,c, we illustrate the transformation of the DCMST instance into a BLMST instance. Here, we first assign a unique label to each node (e.g., labels {a, b, c, d, e}). Then, each edge inherits labels from the nodes it connects, effectively defining its own set of labels. The budget in the BLMST instance is set for each label, which corresponds to the degree constraint of $k=2$ from the DCMST problem. Thus, the budget for each label, pertaining to the edges labeled with it, is also $k=2$. Figure 1d shows a spanning tree constituting the optimal solution of both the DCMST and BLMST instances. Given the constraints and cost assumptions, the same spanning tree is identified as the optimal solution for both problems, demonstrating the effectiveness of the reduction.

5. Mathematical Formulations

The classical packing formulation based on the well-known subtour elimination constraints [29] can be extended to impose the budget requirements. The resulting linear program (LP) is as follows.

$$\begin{array}{ccc}\hfill \underset{x}{min}\sum _{\{i,j\}\in E}& c_{ij}{x}_{ij}\hfill & \hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.\sum _{\{i,j\}\in E}{x}_{ij}& =\left|V\right|-1\hfill & \hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill \sum _{j\in \delta \left(i\right)}{x}_{ij}& \ge 1\hfill & \forall i\in V\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \sum _{\{i,j\}\in E\left(S\right)}{x}_{ij}& \le \left|S\right|-1\hfill & \forall S\subset V:\left|S\right|\ge 3\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill \sum _{\{i,j\}\in E_{\ell }}{x}_{ij}& \le B_{\ell }\hfill & \forall \ell \in L\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill x_{ij}& \in \{0,1\}\hfill & \forall \{i,j\}\in E\hfill \end{array}$$

(6)

Formulation (1)–(6) uses a binary decision variable $x_{ij}$ associated with each edge $\{i,j\}\in E$, defined in such a way that $x_{ij}$ assumes a value equal to 1 if $\{i,j\}$ belongs to the spanning tree, and 0 otherwise. The objective function (1) minimizes the total cost associated with the selected edges, i.e., the ones s.t. $x_{ij}=1$. In order to obtain a tree, constraints (2) impose that the number of selected edges is exactly equal to the number of vertices minus one, while constraints (4) prevent cycles by forbidding, for each subset of nodes $S\subset V$, the selection of strictly more than $\left|S\right|-1$ edges with both endpoints in S. Furthermore, constraints (3) are valid inequalities, satisfied by every feasible solution of the BLMST problem, eventually improving the continuous relaxation of Formulation (1)–(6) by requiring that each vertex is spanned by at least one edge. Finally, constraints (5) impose, for each label $\ell \in L$, that the number of selected edges associated with such a label, i.e., the number of edges $\{i,j\}$ in $E_{\ell }$ s.t. $x_{ij}=1$, does not exceed the budget available $B_{\ell }$.

Alternatively, constraints (5) can be modified by using additional variables to keep track of budget violations and assigning sufficiently large cost coefficients to those variables, actually preventing budget violations. To this aim, let us introduce a binary auxiliary variable $y_{\ell }$ defined for each $\ell \in L$, equal to 1 if the number of edges $\{i,j\}\in E$ s.t. $\ell \in \varphi \left(\right\{i,j\left\}\right)$ and $x_{ij}=1$ is strictly larger than $B_{\ell }$. The resulting penalization-based formulation is given below.

$$\begin{array}{ccc}\hfill \underset{x}{min}\sum _{\{i,j\}\in E}{c}_{ij}{x}_{ij}& +\mathcal{P}\sum _{\ell \in L}{y}_{\ell }\hfill & \hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.\sum _{\{i,j\}\in E}{x}_{ij}& =\left|V\right|-1\hfill & \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \sum _{j\in \delta \left(i\right)}{x}_{ij}& \ge 1\hfill & \forall i\in V\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \sum _{\{i,j\}\in E\left(S\right)}{x}_{ij}& \le \left|S\right|-1\hfill & \forall S\subset V:\left|S\right|\ge 3\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill \sum _{\{i,j\}\in E_{\ell }}{x}_{ij}& \le |E_{\ell }|y_{\ell }+B_{\ell }\hfill & \forall \ell \in L\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill x_{ij}& \in \{0,1\}\hfill & \forall \{i,j\}\in E\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill y_{\ell }& \in \{0,1\}\hfill & \forall \ell \in L\hfill \end{array}$$

(13)

The objective function (7) sums up the costs associated with the selected edges and a penalty $\mathcal{P}$ for each budget violation. The value of $\mathcal{P}$ must be an upper bound on the cost of every feasible solution of the BLMST problem. A natural choice for $\mathcal{P}$ is the cost of the maximum spanning tree of G, which can be computed in polynomial time by replacing the cost of each edge with its additive inverse and then computing the minimum spanning tree of the resulting graph [30].

A third alternative is to incorporate the budget constraints within a single-commodity flow formulation, in which an arbitrary node $s\in V$ is selected as the source node, from which $\left|V\right|-1$ units of flow are sent through the network, each destined for one of the remaining nodes. Continuous variables are associated with each arc of the directed network $G_D=(V,A)$, having the same node set as G and an edge set containing both the arcs $(i,j)$ and $(j,i)$ for every edge $\{i,j\}\in E$. Specifically, a variable $f_{ij}\ge 0$ represents the amount of flow on the arc $(i,j),\forall (i,j)\in A$. If a positive flow is sent either on $(i,j)$ or $(j,i)$, then $\{i,j\}$ belongs to the spanning tree. Below is the compact single-commodity flow formulation.

$$\begin{array}{ccc}\hfill \underset{f,x}{min}\sum _{(i,j)\in A}& c_{ij}{x}_{ij}\hfill & \hfill \end{array}$$

(14)

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

(15)

$$\begin{array}{ccc}\hfill \sum _{(i,j)\in A}{f}_{ij}-\sum _{(j,i)\in A}{f}_{ji}& =\left\{\begin{array}{cc}\left|V\right|-1\hfill & \mathrm{if}i=s\hfill \\ -1\hfill & \mathrm{if}i\in V\setminus \left\{s\right\}\hfill \end{array}\right.\hfill & \forall i\in V\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill f_{ij}& \le \left(\right|V|-1)x_{ij}\hfill & \forall (i,j)\in A\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill x_{ij}+x_{ji}& \le 1\hfill & \forall \{i,j\}\in E\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill \sum _{(i,j)\in A_{\ell }}{x}_{ij}& \le B_{\ell }\hfill & \forall \ell \in L\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill x_{ij}& \in \{0,1\}\hfill & \forall (i,j)\in A\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill f_{ij}& \ge 0\hfill & \forall (i,j)\in A\hfill \end{array}$$

(21)

The objective function (14) minimizes, as before, the total cost of the tree. The selection of exactly $\left|V\right|-1$ arcs is imposed via constraints (15), while constraints (16) are flow conservation constraints guaranteeing that each node receives exactly one unit of flow, except for the source. Constraints (17) link the f and x variables by allowing a positive flow on a given arc only if the corresponding edge belongs to the spanning tree. Although the model contains two variables for each edge of the original graph, constraints (18) prevent the related directed arcs from simultaneously carrying positive flow. Finally, the budget restrictions are imposed via constraints (19), where $A_{\ell }$ denotes the set of directed arcs of $G_D$ whose corresponding edges of G have label ℓ, i.e., $A_{\ell }=\{(i,j)\in A:\ell \in \varphi \left(\{i,j\}\right)\}$.

6. Properties of BLMST Solutions

Since, without the restriction on the budget, the BLMST problem reduces to the MST problem, the cost associated with an optimal solution of the latter constitutes a trivial lower bound for the former, as no spanning tree of G of a smaller cost exists. Furthermore, let us denote by $\varphi \left(i\right)$ the set of labels associated with at least one edge incident on node i, $\forall i\in V$.

$$\varphi \left(i\right)=\bigcup _{j\in \delta \left(i\right)}\varphi \left(\{i,j\}\right)$$

(22)

In any feasible solution of the BLMST, the number of selected edges incident on any node i is always at least one, for node i to the spanned, and never exceeds the sum of the budgets available for the labels associated with such edges.

$$\begin{array}{ccc}\hfill & 1\le \sum _{j\in \delta \left(i\right)}{x}_{ij}\le \sum _{\ell \in \varphi \left(i\right)}{B}_{\ell }\hfill & \forall i\in V\hfill \end{array}$$

(23)

7. Lagrangian Approach

To provide lower bounds for the BLMST problem, we employed a methodology inspired by the Lagrangian Relaxation method, previously adopted in other MST problem extensions [27,31], as well as in other contexts including assignment, network optimization, and wireless sensor network problems [32]. This involves the creation of a Lagrangian relaxation for the problem and the resolution of its associated Lagrangian dual. In our strategy, we include a greedy heuristic at the start of the Lagrangian dual algorithm to potentially enhance the upper bound for the BLMST problem.

7.1. Greedy Algorithm for BLMST

In the following, a heuristic to find feasible solutions to the problem is described.

Algorithm 1 is a modification of Kruskal’s algorithm, adapted for the BLMST problem. It begins by initializing an empty tree (Line 1) and setting the residual budget for each label (Line 2). The edges of the graph G are sorted in non-decreasing order of cost (Line 3). The algorithm then iteratively examines each edge $\{i,j\}\in E$ (Line 4), verifying whether adding it to the tree would violate the budget constraints of any label, while also ensuring that it does not introduce cycles (Line 5). If the edge can be added without violating these constraints, it is added to the tree (Line 6) and the residual budgets are updated accordingly (Line 7). The process continues until all edges have been examined. The algorithm returns the constructed spanning tree (Line 10). Due to the budget constraints, the algorithm cannot guarantee a feasible solution as Kruskal’s algorithm does. Consequently, the resulting solution might be disconnected.

Algorithm 1: GreedyBLMST

7.2. Lagrangian Relaxation

We address the BLMST problem by focusing on the classical packing formulation introduced in Section 5. This formulation serves as the foundation for applying the Lagrangian relaxation method, allowing us to relax the budget constraints (5) and integrate them into the objective function. The relaxation process involves introducing a Lagrangian multiplier ${\lambda }_{\ell }\ge 0$ for each label $\ell \in L$, which leads to our Lagrangian function $\mathcal{L}(x,\lambda )$. Each multiplier can be seen as a “tool” for weighing the importance of the associated constraint in the objective function, allowing us to remove the constraints in the definition of the feasible region of the relaxed problem, but taking into account the satisfaction or violation of the constraints themselves in the objective function. The term added to the original objective function rewards the satisfaction of the constraints and penalizes their violation, so the objective function goes in the direction of satisfying all constraints. If the multiplier is too small, the associated constraint’s impact on the objective function becomes minimal, leading to insufficient emphasis on fulfilling this constraint. Conversely, if the multiplier is too large, it can overly influence the objective function, causing it to focus excessively on satisfying the particular constraint. It is not trivial to find the right values to associate with these multipliers. This function modifies the original objective by integrating the weighted relaxed constraints, resulting in

$$\begin{array}{c}\hfill \mathcal{L}(x,\lambda )=\sum _{\left\{ij\right\}\in E}{c}_{ij}{x}_{ij}-\sum _{\ell \in L}\left({\lambda }_{\ell }{B}_{\ell }\right)+\sum _{\ell \in L}\sum _{\left\{ij\right\}\in E_{\ell }}{\lambda }_{\ell }{x}_{ij}\end{array}$$

which leads us to formulate the Relaxed Lagrangian Problem (RLP) as follows.

$$\begin{array}{ccccc}\hfill RLP& \mathcal{L}(x,\lambda )\hfill & \hfill =\sum _{\{i,j\}\in E}{c}_{ij}{x}_{ij}& -\sum _{\ell \in L}\left({\lambda }_{\ell }{B}_{\ell }\right)+\sum _{\ell \in L}\sum _{\{i,j\}\in E_{\ell }}{\lambda }_{\ell }{x}_{ij}\hfill & \end{array}$$

(24a)

$$\begin{array}{ccccc}\hfill & \hfill & \hfill \mathrm{s}.\mathrm{t}.\sum _{\{i,j\}\in E}{x}_{ij}& =\left|V\right|-1\hfill & \hfill \end{array}$$

(24b)

$$\begin{array}{ccccc}\hfill & \hfill & \hfill \sum _{j\in \delta \left(i\right)}{x}_{ij}& \ge 1\hfill & \forall i\in V\hfill \end{array}$$

(24c)

$$\begin{array}{ccccc}\hfill & \hfill & \hfill \sum _{\{i,j\}\in E\left(S\right)}{x}_{ij}& \le \left|S\right|-1\hfill & \forall S\subset V:\left|S\right|\ge 3\hfill \end{array}$$

(24d)

$$\begin{array}{ccccc}\hfill & \hfill & \hfill x_{ij}& \in \{0,1\}\hfill & \forall \{i,j\}\in E\hfill \end{array}$$

(24e)

Except for the constant term $-{\sum }_{\ell \in L}\left({\lambda }_{\ell }{B}_{\ell }\right)$, Formulation (24) essentially conforms to a standard MST problem. In this scenario, the edge weights are determined according to a specific assignment rule: $\widehat{c_{ij}}=c_{ij}+{\sum }_{\ell \in \varphi \left(\right\{i,j\left\}\right)}{\lambda }_{\ell }$. Given this property, the RLP can be solved using a classic greedy algorithm typically designed for the MST problem (like Kruskal, Prim). The feasible set of the RLP is defined as   

$$\begin{array}{c}\hfill \mathcal{X}=\left\{x_{ij}\in {\{0,1\}}^{\left|E\right|}:\sum _{\{i,j\}\in E}{x}_{ij}=\left|V\right|-1,\sum _{\left\{ij\right\}\in E\left(S\right)}{x}_{ij}\le \left|S\right|-1\forall S\subset V,\left|S\right|\ge 3\right\}\end{array}$$

thus, we can reformulate the RLP in a compact way:

$$\begin{array}{c}\hfill RLP:\underset{x\in \mathcal{X},\lambda \ge 0}{min}\mathcal{L}(x,\lambda )\end{array}$$

(25)

For a given vector $\lambda$ of Lagrangian multipliers and a corresponding set of decision variables x for the problem, we define an instance of RLP as $RLP(\lambda ,x)$. The Lagrangian Dual is formulated as

$$\begin{array}{c}\hfill LD:\underset{\lambda \ge 0}{max}\underset{x\in \mathcal{X}}{min}\mathcal{L}(x,\lambda )\end{array}$$

(26)

In addressing the Lagrangian dual for this problem, we encounter a maximization problem where the objective function exhibits concavity and a piecewise affine nature, indicative of its nonsmooth characteristic. A comprehensive exploration of both theoretical foundations and practical computational strategies for nonsmooth optimization can be found in [33]. The LD problem could be solved as a Continuous Linear Programming problem, but it is often very computationally expensive. To solve LD, other heuristic methods are therefore used, including the Subgradient algorithm which is a technique for determining good Lagrangian multipliers.

7.3. Subgradient Algorithm for the BLMST Problem

We now proceed to detail our approach for updating multipliers in tackling the Lagrangian dual. In this context, we delve into the dual ascent strategy, a technique for handling nonsmooth optimization, which is crucial for enhancing our approach to derive effective lower bounds while maintaining computational efficiency.

The Subgradient Algorithm for BLMST (Algorithm 2) requires the following inputs: (i) a graph G, which serves as the problem’s structure; (ii) the labeling function $\varphi$; (iii) $\alpha$, a parameter influencing the rate of change in the Lagrangian multipliers; (iv) $\Delta$, the iteration threshold, which is an integer determining when to modify the step size; (v) $\Omega$, the maximum number of iterations the algorithm will perform; and (vi) $\epsilon$, the convergence tolerance, representing a stopping criterion of the algorithm based on the convergence toward the lower bound.

The algorithm outputs the tighter lower bound identified for the BLMST problem. During the initialization phase (Lines 1–7), the algorithm sets the initial values of the Lagrangian multipliers $\left({\lambda }_{\ell }\right)$ equal to zero. Then, it sets the lower bounds $LB$ and $\overline{LB}$ to negative infinity, where $LB$ records the current lower bound, while $\overline{LB}$ tracks the previously computed lower bound. After that, the algorithm initializes the upper bound $UB$ using the UpperBoundInitialization procedure. This procedure returns the value of the solution computed via the GreedyBLMST algorithm when feasible; otherwise, it calculates the value of the maximum MST. At the end of the initialization phase, the algorithm initializes the iteration counter i and the improvement iteration tracker $imprIteration$. The counter i monitors the current iteration number, while $imprIteration$ keeps track of the last iteration in which the bound was updated.

The iteration phase of the algorithm starts at Line 8 and consists of a singular while loop spanning Lines 9–23. Each iteration involves updating the vector $\lambda$ to identify the most effective Lagrangian multipliers. The loop iterates up to a maximum of $\Omega$ times. Delving into the procedural steps within each iteration, at Line 10, the algorithm starts the construction of the Relaxed Lagrangian Problem (RLP), using the current values of the $\lambda$ multipliers and the x variables, extracted from the graph G; the function $\varphi$ is used to construct the budget constraints. At Line 11, the algorithm optimally solves the subproblem, yielding the optimal solution $x^{*}$ for the relaxed Lagrangian. On Lines 12–14, the algorithm evaluates whether the value of the Lagrangian function $\mathcal{L}(x^{*},\lambda )$, calculated for the current iteration, exceeds the previously established lower bound $LB$. If so, the algorithm updates $LB$ and stores the current iteration into $imprIteration$. The algorithm performs the computation of the Lagrangian subgradients $s_{\ell }$ at Line 15, subsequently determining the value of the step size parameter $\theta$ at Line 16. As already mentioned, the Lagrangian multipliers provide a reward or a penalty based on whether the solution found satisfies the relaxed constraints or not, pushing the objective function to find a solution that does not violate the relaxed constraints. Here, $s_{\ell }$ indicates the growth direction of the objective function, and $\theta$ represents the length of the step. On Lines 19–20, the algorithm verifies if the lower bound is equal to the upper bound and checks how much the lower bound computed in the current iteration ($LB$) differs from the previous one ($\overline{LB}$). On Line 20, the algorithm updates $\overline{LB}$. The if statement on Lines 21–22 evaluates whether there has been no enhancement in the lower bound over the last $\Delta$ iterations. If this is the case, the value of $\alpha$ is halved to moderate the convergence of the algorithm. On Line 23, the algorithm increases the iteration counter i, while on Line 24, the best computed lower bound is returned.

Algorithm 2: Subgradient algorithm for BLMST

The UpperBoundInitialization algorithm (Algorithm 3) provides an upper bound estimate for the BLMST problem. It first runs the GreedyBLMST algorithm to eventually build a spanning tree T from the given graph G and labeling function $\varphi$ (Line 1). If the resulting tree T is a feasible solution for the BLMST problem, then the algorithm computes and returns the cost associated with T (Line 2), which represents an upper bound for the value of the optimal solution. However, if T is not feasible for the BLMST problem, the algorithm calculates the cost of the Maximum Spanning Tree (MaxMST) of G (Line 6). The cost of such a tree, even if associated with an infeasible solution for BLMST, provides an upper bound for any feasible solution of the problem. The algorithm then returns this value as the upper bound (Line 7). This approach ensures that an upper bound is always provided.

Algorithm 3: UpperBoundInitialization

8. Computational Experiments

In this section, we compare the proposed methods presenting the results of our computational tests. The methods were coded in Python on an OSX platform, running on an Apple M2 Max system with a 64 GB unified memory. All the mathematical formulations were solved by using the IBM ILOG CPLEX 12.1.1 solver with a maximum number of threads equal to three and an imposed time limit of 1800 s.

8.1. Test Instances

Since no benchmark instances for the BLMST problem are available in the literature, we generated a set of random instances according to some choices presented in the following. Provided the desired number of nodes $\left|V\right|$, a uniformly random Prüfer sequence of length $\left|V\right|-2$ is generated and then converted to the unique tree T associated with such sequence. Subsequently, a set of labels L is generated, according to a target size $\left|L\right|\in \{5,10,20\}$. Given a budget $B_{\ell }$ associated with each label $\ell \in L$, ℓ is assigned to exactly $B_{\ell }$ randomly sampled edges of T, which ensures, via construction, the existence of at least one feasible solution of BLMST, i.e., the spanning tree T. A graph $G=(V,E)$ is obtained by iteratively adding edges to T until a target density d is reached, where $d=\frac{2\left|E\right|}{\left|V\right|\left(\right|V|-1)}$. Finally, a random cost $w_{ij}\in [w_{min},w_{max}]$ is assigned to each edge in $\{i,j\}\in E$ and each label is assigned to each edge of G that does not belong to T with probability $\gamma =0.4$. In this way, one or more labels will be assigned to each edge.

We generated 162 instances by setting different values for each parameter. More in detail, in order to further investigate the impact of the budget restrictions, we organized the instances in three scenarios, composed of 54 instances each, depending on the values used for $B_{\ell }$. Fixed budget values have been considered, as detailed below.

Scenario 1:

$B_{\ell }=\lceil \left|V\right|/2\rceil ,\forall \ell \in L$;

Scenario 2:

$B_{\ell }=\lceil \left|V\right|/3\rceil ,\forall \ell \in L$;

Scenario 3:

$B_{\ell }=\lceil \left|V\right|/4\rceil ,\forall \ell \in L$.

The tighter the budget, the more challenging is expected to be the resolution of an instance; thus, we will compare the performances of the proposed methods as $B_{\ell }$ varies.

8.2. Results Discussion

In this section, we present the computational results obtained from solving the mathematical models using the commercial solver CPLEX, as well as the results obtained with our Subgradient algorithm.

8.2.1. Mathematical Models

We start our discussion with a comparison of the three formulations for Scenario 1. The results of this comparison are contained in

Table 1. Comparison of the three mathematical formulations on the set of instances with budget fixed to $\lceil \left|V\right|/2\rceil$ (Scenario 1), in terms of optimal solutions, computational time, and nodes of the B&B tree.

			Packing Formulation			Penalization-Based Formulation			Single-Commodity Flow Formulation
L	n	d	Obj	Nodes	Time	Obj	Nodes	Time	Obj	Nodes	Time
5	50	0.2	384	6782	1.66	384	6868	2.01	384	0	0.11
		0.3	207	3	0.18	207	3	0.19	207	548	0.12
		0.4	137	3	0.17	137	3	0.17	137	220	0.08
	100	0.2	338	170,424	50.84	338	143,693	43.92	338	16,392	2.84
		0.3	247	5044	3.20	247	4624	3.63	247	18,141	4.69
		0.4	223	45,068	24.33	223	136,486	79.59	223	196,435	36.04
	150	0.2	372(u)	966,652	1802.89	372(u)	1,193,242	1800.89	372	1,092,872	43.13
		0.3	278	9496	8.71	278	21,913	17.00	278	1244	0.74
		0.4	218(u)	879,093	1803.81	218(u)	933,306	1804.57	218	3,797,326	1531.55
	200	0.2	398(u)	963,689	1804.88	398(u)	1,057,325	1804.26	398	4820	3.12
		0.3	283(u)	1,219,447	1800.80	283(u)	1,276,533	1800.75	283	678,958	63.50
		0.4	241(u)	792,379	1800.97	241(u)	910,489	1800.82	241	1074	13.16
	250	0.2	445(u)	735,126	1802.51	445(u)	546,438	1803.44	445	4980	29.21
		0.3	353(u)	331,215	1803.87	349(u)	220,756	1805.90	348	802,163	364.94
		0.4	307(u)	424,095	1801.18	307(u)	355,613	1802.86	307	2,434,929	1084.20
	300	0.2	470(u)	541,077	1803.19	471(u)	443,603	1803.94	470(u)	1,293,501	1801.22
		0.3	365(u)	155,207	1806.46	365(u)	197,361	1805.92	364(u)	221,665	1800.01
		0.4	331(u)	312,917	1801.73	331(u)	310,421	1801.46	331	2783	5.45
		Avg	310.94	419,873.17	1106.74	310.61	431,037.61	1110.07	310.61	587,113.94	376.89
		#Opt	7			7			16
10	50	0.2	269	41	0.22	269	42	0.20	269	88	0.06
		0.3	214	46	0.20	214	124	0.30	214	66	0.05
		0.4	199	0	0.13	199	0	0.06	199	0	0.08
	100	0.2	368	284	1.13	368	518	1.32	368	1838	0.59
		0.3	233	18,749	8.84	233	12,504	6.75	233	2294	1.96
		0.4	192	154	0.96	192	108	0.99	192	1,983,767	155.06
	150	0.2	368(u)	708,360	1804.84	368(u)	744,156	1805.48	368(u)	2,981,555	1800.00
		0.3	278(u)	1,013,942	1801.29	278(u)	1,946,679	1801.66	278	24	0.67
		0.4	239(u)	1,149,943	1802.87	239(u)	1,056,962	1803.49	239	2246	1.33
	200	0.2	413(u)	2,079,434	1800.00	413(u)	2,093,693	1800.01	413	674,995	104.59
		0.3	333(u)	388,022	1805.55	333(u)	831,062	1803.80	333	2134	1.77
		0.4	271(u)	240,934	1806.18	271(u)	346,307	1805.58	271	1,099,752	388.40
	250	0.2	434(u)	314,789	1805.58	434(u)	432,382	1805.70	434(u)	5,105,688	1800.01
		0.3	368(u)	363,941	1804.82	368(u)	346,884	1806.18	368(u)	1,627,238	1801.54
		0.4	301(u)	400,755	1802.14	301(u)	431,307	1801.48	301	7071	107.48
	300	0.2	439(u)	902,884	1801.19	439(u)	347,996	1803.33	439(u)	728,677	1800.62
		0.3	372(u)	370,704	1802.33	372(u)	324,579	1801.83	372(u)	565,164	1800.88
		0.4	340(u)	307,286	1801.38	341(u)	283,720	1802.40	340	13	4.51
		Avg	312.83	458,903.78	1202.76	312.89	511,056.83	1202.81	312.83	821,256.11	542.76
		#Opt	6			6			13
20	50	0.2	322	2098	0.71	322	2316	0.95	322	3269	0.27
		0.3	200	4131	1.08	200	3115	0.81	200	541	0.13
		0.4	145	60	0.24	145	74	0.24	145	883	0.34
	100	0.2	327	1401	1.71	327	1305	1.77	327	42,280	8.21
		0.3	219(u)	952,555	1803.03	219(u)	845,953	1802.63	219	1563	2.02
		0.4	192(u)	2,739,262	1800.00	192(u)	3,288,212	1800.00	192	6	0.30
	150	0.2	405	1,762,851	1075.32	405	2,079,591	1365.48	405	0	0.62
		0.3	268	1,260,067	923.22	268	1,295,691	943.48	268(u)	15,031,433	1800.01
		0.4	214(u)	1,257,318	1801.46	214(u)	998,077	1804.00	214(u)	6,895,492	1800.01
	200	0.2	411(u)	1,314,070	1803.33	411(u)	960,992	1803.75	411(u)	1,913,636	1800.01
		0.3	302(u)	1,101,446	1801.22	302(u)	977,988	1801.71	302(u)	1,341,987	1801.07
		0.4	266(u)	642,759	1802.37	266(u)	656,149	1802.80	266	12,797	57.58
	250	0.2	408(u)	713,336	1803.55	408(u)	627,621	1804.91	408(u)	805,348	1801.46
		0.3	333	420	7.88	333	1902	12.52	333(u)	374,960	1801.52
		0.4	286(u)	467,169	1801.14	286(u)	457,607	1801.27	286	10,162	83.79
	300	0.2	442(u)	310,257	1802.56	442(u)	279,823	1803.88	441(u)	481,375	1800.85
		0.3	351(u)	344,467	1801.79	351(u)	385,789	1801.30	351	54,333	29.49
		0.4	321(u)	280,710	1802.75	321(u)	277,663	1803.19	321	22,376	294.25
		Avg	300.67	730,798.72	1212.97	300.67	729,992.67	1230.82	300.61	1,499,580.06	726.77
		#Opt	7			7			11

in which the first three columns report the information about the instances: the number of labels $\left|L\right|$, the number of nodes ($\left|V\right|$), and the network density (d). When available, the fourth column, heading Obj, reports the value of the optimal solution. If no optimal solution is available, (u) appears close to the value of the best feasible solution computed within the time limit. If no solution is available, (l) appears close to the value of the best lower bound given by the formulation. The fifth column, named Nodes, reports the number of nodes explored by CPLEX in the B&B tree, while the sixth column, i.e., Time, reports the computational times in seconds. The fourth, fifth, and sixth columns refer to the results obtained by Formulation (1)–(6), while the subsequent columns report the same information for Formulations (7)–(13) and (14)–(21), respectively.

The content of the table is divided into three main groups of instances, sharing the same number of labels $\left|L\right|\in \{5,10,20\}$. For each of these groups, two additional rows report the number of optimal solutions found by each formulation (#Opt), along with the average cost of the identified solution, computed considering only the instances where at least an upper bound is available, the average number of nodes of the B&B tree, and the average run time in seconds. The results show the computational dominance, on the tested instances, of the single commodity flow formulation, which optimally solves 40 out of 54 instances of Scenario 1, while both the packing- and the penalization-based formulations certify the optimality of the identified solution for only half this number of instances. Despite this, focusing on the upper bound values, we may notice that only in three cases the packing formulation identifies a worse solution than the single commodity flow formulation (i.e., scenarios 5-250-0.3, 5-300-0.3, and 20-300-0.2, where the notation $\left|L\right|$-$\left|V\right|$-d indicates an instance with $\left|L\right|$ labels, $\left|V\right|$ nodes, and a network density equal to d) and the cost percentage gap with respect to the solution identified by the latter is always reduced (1.44%, 0.27%, and 0.23%, respectively). On the other hand, the penalization-based formulation records five positive cost percentage gap values (i.e., scenarios 5-250-0.3, 5-300-0.2, 5-300-0.3, 10-300-0.4, and 20-300-0.2), but the maximum gap value equals 0.29%.

Table 2. Comparison of the three mathematical formulations on the set of instances with budget fixed to $\lceil \left|V\right|/3\rceil$ (Scenario 2), in terms of optimal solutions, computational time, and nodes of the B&B tree.

			Packing Formulation			Penalization-Based Formulation			Single-Commodity Flow Formulation
L	n	d	Obj	Nodes	Time	Obj	Nodes	Time	Obj	Nodes	Time
5	50	0.2	386	386	0.31	386	420	0.39	386	1438	0.25
		0.3	280	284	0.25	280	218	0.27	280	2969	0.51
		0.4	199	8356	1.86	199	19,979	4.21	199	184	0.18
	100	0.2	403(u)	1,542,399	1800.47	403(u)	1,836,446	1800.02	402	4543	2.18
		0.3	300	617,286	285.06	300	612,969	324.27	300	9049	8.58
		0.4	209	224	1.01	209	107	1.04	209	787	1.96
	150	0.2	439(u)	1,332,880	1801.59	439(u)	1,574,331	1800.95	438	45,539	24.32
		0.3	299(u)	2,003,733	1800.98	299(u)	1,926,242	1800.01	299(u)	3,791,744	1800.01
		0.4	249(u)	822,090	1802.28	249(u)	915,049	1801.69	249	3966	17.59
	200	0.2	444(u)	892,403	1801.70	444(u)	859,623	1801.52	442	23,864	44.95
		0.3	344(u)	425,359	1803.16	341(u)	638,421	1801.35	339	723,895	563.11
		0.4	275(u)	621,466	1801.22	271(u)	619,923	1800.90	269	302,171	275.51
	250	0.2	476(u)	564,203	1802.37	476(u)	433,577	1802.12	473	397,170	185.26
		0.3	346(u)	585,841	1800.88	343(u)	554,115	1801.48	343	17,252	112.88
		0.4	305(u)	366,937	1801.10	305(u)	408,721	1800.87	303	12,448	249.95
	300	0.2	512(u)	430,704	1802.51	513(u)	479,703	1801.36	508	11,792	149.76
		0.3	387(u)	411,333	1801.03	383(u)	449,827	1800.72	381	685,217	534.54
		0.4	347(u)	230,884	1801.53	342(u)	252,950	1801.57	342	6387	186.97
		Avg	344.44	603,153.78	1317.18	310.61	643,478.94	1319.15	342.33	335,578.61	231.03
		#Opt	5			5			17
10	50	0.2	443	1452	0.40	443	2102	0.55	443	1435	0.28
		0.3	224	15	0.21	224	13	0.27	224	381	0.14
		0.4	214	1948	0.71	214	2742	0.74	214	8402	2.00
	100	0.2	444	39,280	16.13	444	50,051	15.27	444	6364	2.17
		0.3	287	21,886	8.84	287	11,409	5.99	287	33,229	14.82
		0.4	208	103,006	61.86	208	294,845	204.59	208	2737	4.15
	150	0.2	487(u)	660,232	2158.41	484(u)	791,291	2126.62	476	31,726	33.58
		0.3	339(u)	931,630	1800.99	334(u)	1,401,348	1800.88	332	3749	6.83
		0.4	238(u)	699,323	1801.22	238(u)	623,122	1801.58	233	5176	42.35
	200	0.2	438(u)	1,129,062	1801.01	435	1,056,214	1384.73	435	182,707	106.78
		0.3	326(u)	806,011	1800.68	323(u)	817,200	1800.87	322	9207	72.55
		0.4	276(u)	662,800	1800.71	276(u)	661,045	1800.79	274	42,928	450.91
	250	0.2	477(u)	566,240	1801.37	479(u)	388,271	1801.30	471(u)	245,438	1800.01
		0.3	386(u)	570,496	1800.55	392(u)	416,965	1801.22	383	97,664	102.45
		0.4	308(u)	408,740	1800.76	311(u)	352,942	1801.15	304	251,202	483.13
	300	0.2	492(u)	477,226	1800.96	491(u)	397,172	1801.37	487	95,139	257.33
		0.3	410(u)	346,901	1800.80	413(u)	315,959	1800.02	405	42,303	266.19
		0.4	355(u)	244,217	1800.94	349(u)	274,325	1800.64	345	15,707	590.32
		Avg	352.89	426,136.94	1225.36	352.50	436,500.89	1208.26	349.28	59,749.67	235.33
		#Opt	6			7			17
20	50	0.2	521	4745	0.72	521	5333	0.85	521	8961	1.87
		0.3	422	193,353	30.83	422	135,239	21.36	422	235,328	87.73
		0.4	289	33,520	6.58	289	39,737	7.48	289	2985	1.56
	100	0.2	450(u)	1,482,149	1800.56	452(u)	1,735,218	1800.31	436(u)	1,806,597	1800.03
		0.3	370(u)	2,053,794	1800.66	375(u)	1,586,947	1800.83	372(u)	1,140,036	1800.93
		0.4	259	175,896	109.85	259	322,529	182.48	259	39,805	23.34
	150	0.2	473(u)	1,274,220	1800.95	468(u)	1,411,355	1800.67	465(u)	562,841	1800.52
		0.3	374(u)	718,963	1801.45	374(u)	872,929	1801.20	360	509,112	519.49
		0.4	293(u)	947,147	1800.68	291(u)	1,111,344	1800.64	286(u)	283,493	1800.01
	200	0.2	547(u)	269,533	1801.91	534(u)	360,040	2082.81	516(u)	223,098	1800.00
		0.3	428(u)	390,888	2090.58	443(u)	332,352	2070.14	411(u)	129,086	1800.32
		0.4	331(u)	393,927	1800.96	322(u)	568,058	1800.58	316(u)	132,820	1800.01
	250	0.2	639(u)	218,842	1801.61	563(l)	218,550	1801.24	583(u)	82,590	1800.57
		0.3	435(u)	284,770	1801.05	425(u)	343,136	1800.85	406(l)	54,569	1800.23
		0.4	357(u)	303,947	1800.88	343(l)	194,942	1801.34	342(l)	54,360	1800.01
	300	0.2	550(u)	258,326	1800.97	519(l)	179,232	1801.44	528(u)	108,351	1800.01
		0.3	453(l)	167,726	1800.85	454(l)	168,103	1801.02	455(l)	49,537	1800.01
		0.4	360(u)	159,818	1800.77	333(l)	150,825	1800.76	335(l)	32,657	1800.02
		Avg	396.42	661,511.25	1237.14	395.83	706,756.75	1264.11	387.75	422,846.83	1102.99
		#Opt	5			9			9

and

Table 3. Comparison of the three mathematical formulations on the set of instances with budget fixed to $\lceil \left|V\right|/4\rceil$ (Scenario 3), in terms of optimal solutions, computational time, and nodes of the B&B tree.

			Packing Formulation			Penalization-Based Formulation			Single-Commodity Flow Formulation
L	n	d	Obj	Nodes	Time	Obj	Nodes	Time	Obj	Nodes	Time
5	50	0.2	454	0	0.11	454	0	0.06	454	0	0.05
		0.3	261	0	0.12	261	5	0.06	261	448	0.20
		0.4	250	0	0.06	250	0	0.06	250	0	0.07
	100	0.2	401	191,501	81.85	401	248,154	98.99	401	1696	1.25
		0.3	333	2957	2.49	333	6357	4.81	333	2052	2.53
		0.4	245(u)	2,059,414	1800.00	245(u)	1,823,294	1800.91	245	2790	4.16
	150	0.2	509(u)	1,360,526	1801.92	507(u)	1,487,666	1801.23	505	4653	6.52
		0.3	342	980,506	929.03	343(u)	1,667,107	1800.89	342	3253	11.80
		0.4	286(u)	1,092,273	1801.61	286(u)	1,250,047	1801.08	286	3450	16.74
	200	0.2	460	305,216	377.34	460	430,473	429.30	460	6097	20.92
		0.3	365(u)	826,199	1801.41	363	426,213	824.94	363	5450	34.37
		0.4	307(u)	604,130	1801.78	311(u)	493,245	1801.52	307	1397	18.61
	250	0.2	577(u)	476,599	1802.55	584(u)	398,520	1801.78	570	81,652	112.37
		0.3	398(u)	555,892	1801.04	398(u)	534,605	1800.91	398	2,591,989	1343.45
		0.4	347(u)	254,453	1801.33	346(u)	311,367	1800.02	342	16,484	236.83
	300	0.2	611(u)	4,32,514	1801.81	608(u)	511,274	1801.26	603	88,904	132.43
		0.3	483(u)	336,811	1801.17	479(u)	365,249	1800.95	476(u)	536,136	1800.02
		0.4	399(u)	186,216	1802.00	399(u)	268,255	1800.80	393	16,441	566.36
		Avg	390.44	536,955.94	1178.20	310.61	567,879.50	1176.09	388.28	186,827.33	239.37
		#Opt	7			7			17
10	50	0.2	577	1303	0.43	577	817	0.31	577	609	0.20
		0.3	392	6487	1.40	392	5872	1.68	392	3116	1.16
		0.4	313	46,141	11.27	313	23,774	5.48	313	1301	0.71
	100	0.2	573	110,883	46.85	573	67,719	22.66	573	2,023,550	1231.06
		0.3	425	26,833	14.15	425	74,065	37.54	425	186,271	77.61
		0.4	321	81,184	50.57	321	572,480	307.12	321	7514	9.04
	150	0.2	681(u)	964,797	1801.54	680(u)	1,444,185	1800.91	678	15,435	48.90
		0.3	433	645,205	1051.72	434(u)	981,298	1800.34	433	4191	21.59
		0.4	328(u)	1,191,207	1800.01	327	998,393	1336.17	327	11,141	17.15
	200	0.2	642(u)	700,386	1801.99	641(u)	570,778	1801.61	629(u)	209,609	1800.03
		0.3	506(u)	346,800	1801.86	501(u)	463,195	1801.27	493	16,151	42.15
		0.4	420(u)	523,715	1800.87	419(u)	585,661	1801.08	415	78,226	196.53
	250	0.2	778(u)	575,043	1801.35	770(u)	595,381	1801.27	762(u)	187,731	1800.01
		0.3	549(u)	456,530	1800.68	546(u)	435,876	1800.63	536	296,875	414.84
		0.4	473(u)	235,167	1801.54	467(u)	221,101	1801.26	458(u)	318,739	1800.04
	300	0.2	753(u)	214,394	1801.70	756(u)	188,903	1801.91	728(u)	115,937	1800.41
		0.3	524(u)	297,710	1800.74	519(u)	351,196	1800.93	514(u)	163,162	1800.02
		0.4	473(u)	109,761	1801.58	464(u)	215,895	1801.14	456(u)	294,166	1800.01
		Avg	508.94	362,974.78	1166.13	506.94	433,143.83	1195.74	501.67	218,540.22	714.53
		#Opt	7			7			12
20	50	0.2	785	378,694	62.50	785	356,543	56.29	785	68,050	19.92
		0.3	633	24,110	3.52	633	11,997	2.09	633	3124	1.66
		0.4	406	125,520	26.92	406	35,563	9.85	406	4559	2.68
	100	0.2	814(u)	2,349,753	1800.00	802(u)	1,515,358	1800.00	794	1,505,384	1319.16
		0.3	636(u)	1,733,329	1801.05	626(u)	1,565,379	1801.35	627(u)	757,130	1800.63
		0.4	451(u)	1,262,717	1800.87	457(u)	1,447,232	1800.74	443	2,064,088	918.08
	150	0.2	977(u)	479,199	1801.68	1033(u)	546,663	1801.87	972(u)	628,408	1800.01
		0.3	702(u)	390,743	1801.29	701(u)	469,899	1801.69	681(u)	110,816	1800.01
		0.4	723(u)	613,482	1801.02	733(u)	598,619	1801.23	720(u)	138,218	1800.01
	200	0.2	1053(l)	234,790	1801.72	1053(l)	249,211	1801.12	1054(l)	135,933	1800.01
		0.3	813(u)	340,702	1801.76	823(u)	418,977	1801.04	754(l)	72,420	1800.01
		0.4	543(l)	210,307	1801.73	631(u)	322,844	1801.55	545(l)	65,868	1800.01
	250	0.2	1063(l)	206,209	1833.14	1300(u)	355,789	1802.19	1071(l)	79,402	1800.01
		0.3	804(u)	203,529	1801.10	978(u)	207,412	1801.59	729(l)	65,211	1800.03
		0.4	656(l)	187,787	1801.20	810(u)	250,000	1801.32	658(l)	40,527	1800.03
	300	0.2	1316(u)	205,389	1801.07	1242(u)	265,280	1801.10	1121(l)	60,951	1800.01
		0.3	792(l)	169,973	1801.19	979(u)	233,243	1801.26	793(l)	32,974	1800.03
		0.4	635(l)	150,274	1800.60	636(l)	158,053	1800.82	640(l)	27,899	1800.02
		Avg	680.78	817,505.22	1210.98	686.22	727,472.56	1208.35	673.44	586,641.89	1051.35
		#Opt	9			5			14

confirm these results as the imposed budget changes. In particular, on Scenario 2, the single commodity flow formulation identifies 43 out of 54 optimal solutions, while the packing formulation and the penalization-based ones were able to certify optimality only in 16 and 21 cases, respectively. For the first two groups of instances, all the formulations always provided at least an upper bound, while for $\left|L\right|=20$, there exist instances for which CPLEX reaches the imposed time limit without identifying any feasible solution. This occurs in four cases for the single commodity flow formulation, five cases for the penalization-based formulation, and only one case for the packing formulation, which is then the one finding the largest number of feasible solutions in this scenario. Focusing on the instances for which a feasible solution has been identified by all the formulations, the single commodity flow formulation always identifies the best quality solutions, except for one instance (20-100-0.3), where a better solution is identified by the packing formulation and the single commodity flow one exhibits a cost percentage gap of 0.54%. On Scenario 3, the three formulations solve 23, 19, and 43 out of 54 instances to optimality, respectively. Also in this case, the single commodity flow formulation fails to identify a feasible solution for nine instances, while the packing formulation and the penalization-based one provide only lower bounds in six and two cases, respectively. A small positive percentage gap of 0.16%, with respect to the best solution found by the three methods, is recorded for the single commodity flow formulation for only one instance (20-100-0.3).

As we observed, as the number of labels increases and the budget restriction is tighter, the formulations struggle to certify optimality and sometimes even to provide a feasible solution. The single commodity flow formulation generally dominates the other ones, both in terms of computational times and the number of optimally solved instances, although it fails more frequently to identify feasible solutions for $\left|L\right|=20$. In all the scenarios, when reaching the time limit, the solutions identified by the penalized-based formulation are associated with smaller cost percentage gap values with respect to the best than those found by the packing formulation, namely 0.04%, 1.14%, and 0.93% versus 0.02%, 0.94%, and 0.89% in scenarios 1, 2, and 3, respectively.

Restricting the analysis to the best-performing formulation, Figure 2 shows how the number of instances solved to optimality by CPLEX increases with respect to the available computational time. In order to assess also the impact of the number of nodes on the computational times, a curve is associated with each group of instances with different value of $n\in \{50,150,200,250,300\}$, each one including 27 instances. In particular, the x-axis reports the run time limit t (in seconds), while the y-axis reports the number of instances optimally solved within t seconds. As we may notice, the set of instances with $n=50$ is the only one for which all the 27 instances are solved to optimality, and the most challenging instance in this set is solved in ∼100 s. As for the remaining classes, the number of instances solved within 100 s is 20, 16, 11, 2n and 3 for $n=150,200,250,300$, respectively. Overall, the number of instances solved to optimality within the imposed time limit of 1800 s decreases as the value of n increases. With this time limit, indeed, 27 instances with $n=50$, 24 instances with $n=100$, 18 instances with $n=150$, 18 instances with $n=200$, 14 instances with $n=250$, and 12 instances with $n=300$ are solved to optimality.

Let us further investigate the impact of the number of labels on the computational times required by CPLEX to solve Formulation (14)–(21) by looking at the histograms shown in Figure 3. Figure 3a, Figure 3b, and Figure 3c report, for n equal to 200, 250, and 300, respectively, the average run times of CPLEX computed over the instances of each scenario with the same number of labels. As we may notice, considering more labels often results in a relevant increase in the computational times, it is particularly observable when moving from $\left|L\right|=10$ to $\left|L\right|=20$. The only exception is $n=300$ in Scenario 1, where more instances reach the time limit with $\left|L\right|\in \{5,10\}$, thus increasing the average computational times. Tighter budgets also have some impact on the run times, which is generally more detectable for $\left|L\right|=20$.

Finally, let us analyze how the costs of the solutions identified by Formulation (14)–(21) change with respect to the characteristics of the instances. To this aim, we produced three groups of charts, restricting the analysis only to the instances for which at least one feasible solution has been identified by CPLEX. Figure 4 reports costs for instances with five labels. Figure 4a, Figure 4b, and Figure 4c refer to instances with density values equal to $20\%$, $30\%$, and $40\%$, respectively. For the first density class, the cost of the identified spanning tree ranges from 338 associated with $n=100$ in Scenario 1 to 603 associated with $n=300$ in Scenario 3. More in detail, costs in $[338,470]$ were recorded in Scenario 1, costs in $[386,508]$ were recorded in Scenario 2, while costs in $[401,603]$ were recorded in Scenario 3, highlighting higher spanning tree costs associated with tighter budgets. The remaining two density classes exhibit the same trends, with costs ranging from 207 associated with $n=50$ and Scenario 1 to 476 associated with $n=300$ and Scenario 3, in the former case, and from 137 associated with $n=50$ and Scenario 1 to 393 associated with $n=300$ and Scenario 3, in the latter case. From these results, we can further observe a decrease in the spanning tree cost as the density of the network increases, motivated by the availability of a larger number of edges. The charts in Figure 5 and Figure 6 show similar and even more recognizable trends. For $\left|L\right|=10$, costs in $[269,439]$, $[435,387]$, and $[573,762]$ are recorded for Scenarios 1, 2, and 3, respectively, when $d=0.2$; costs in $[214,372]$, $[224,405]$, and $[392,536]$ are recorded for Scenarios 1, 2, and 3, respectively, when $d=0.3$; while costs in $[192,340]$, $[208,345]$, and $[313,458]$ are recorded for Scenarios 1, 2, and 3, respectively, when $d=0.3$. From these charts, we see that the cost difference among the solutions provided for Scenarios 2 and 3 is larger for $d=0.3$ than for $d=0.2$. Finally, for $\left|L\right|=20$, costs in $[322,405]$, $[436,521]$, and $[785,972]$ are recorded for Scenarios 1, 2, and 3, respectively, when $d=0.2$; costs in $[200,268]$, $[360,422]$, and $[627,681]$ are recorded for Scenarios 1, 2, and 3, respectively, when $d=0.3$; while costs in $[192,214]$, $[259,289]$, and $[406,720]$ are recorded for Scenarios 1, 2, and 3, respectively, when $d=0.3$. As we may notice, the cost ranges associated with the three scenarios do not even overlap with each other, making the cost difference observed also for $\left|L\right|\in \{5,10\}$ even more relevant.

8.2.2. Subgradient Algorithm

In our computational experiments, we set the parameters of the Subgradient Algorithm for BLMST as follows: $\alpha =4$, controlling the rate of change in the Lagrangian multipliers; $\Delta =50$, the iteration threshold for modifying the step size; $\Omega =300$, specifying the maximum number of iterations; and $\epsilon =0.05$, the convergence tolerance which serves as the algorithm’s stopping criterion based on proximity to the previous computed lower bound.

Table 4. Upper and lower bounds obtained by running the subgradient algorithm for BLMST (Algorithm 2) on the instances of Scenario 1 (budget fixed to $\lceil \left|V\right|/2\rceil$), Scenario 2 (budget fixed to $\lceil \left|V\right|/3\rceil$), and Scenario 3 (budget fixed to $\lceil \left|V\right|/4\rceil$).

			Scenario 1				Scenario 2				Scenario 3
L	n	d	LB	UB	Time	UB Time	LB	UB	Time	UB Time	LB	UB	Time	UB Time
5	50	0.2	384.00	384(g)	0.007	0.001	380.00	2144	0.007	0.001	408.00	2248	0.094	0
		0.3	207.00	207(g)	0.006	0.001	264.98	2227	0.066	0.001	245.26	2288	0.076	0.001
		0.4	137.00	137(g)	0.007	0.002	189.03	359(g)	0.320	0.002	235.59	2318	0.099	0.001
	100	0.2	338.00	338(g)	0.024	0.005	394.48	4678	0.487	0.001	396.97	4653	0.523	0.001
		0.3	247.00	247(g)	0.035	0.007	285.02	402(g)	0.072	0.009	329.00	4821	1.201	0.002
		0.4	223.00	223(g)	0.046	0.009	202.01	270(g)	0.074	0.01	239.95	4849	1.343	0.002
	150	0.2	372.00	372(g)	0.073	0.015	433.08	7172	2.724	0.003	421.16	922(g)	7.806	0.017
		0.3	278.00	278(g)	0.129	0.021	281.01	402(g)	0.171	0.023	333.49	7304	6.007	0.004
		0.4	218.00	218(g)	0.142	0.028	241.01	386(g)	0.231	0.032	228.90	520(g)	5.143	0.032
	200	0.2	398.00	398(g)	0.185	0.033	421.02	616(g)	0.272	0.051	458.48	9754	14.573	0.004
		0.3	283.00	283(g)	0.247	0.049	320.03	664(g)	0.392	0.055	297.61	776(g)	8.893	0.059
		0.4	241.00	241(g)	0.365	0.064	263.01	442(g)	0.557	0.073	250.10	563(g)	0.760	0.073
	250	0.2	445.00	445(g)	0.317	0.064	471.50	12,273	12.435	0.007	566.03	12,253	26.311	0.007
		0.3	348.00	348(g)	0.490	0.094	337.00	463(g)	0.747	0.111	342.10	785(g)	1.103	0.108
		0.4	307.00	307(g)	0.606	0.124	303.00	392(g)	0.991	0.132	296.04	636(g)	0.955	0.171
	300	0.2	470.00	470(g)	0.550	0.108	490.01	877(g)	0.862	0.118	459.54	1341(g)	37.75	0.123
		0.3	364.00	364(g)	0.814	0.157	369.00	597(g)	1.261	0.169	401.12	1082(g)	1.796	0.183
		0.4	331.00	331(g)	1.062	0.223	342.00	422(g)	1.630	0.222	349.03	793(g)	1.636	0.236
		Avg	310.61	310.61	0.28	0.06	332.62	1932.56	1.29	0.06	347.69	3217.00	6.45	0.06
		#Opt	18				0				0
10	50	0.2	269.00	269(g)	0.005	0.002	405.99	2218	0.075	0.001	521.75	2205	0.201	0.001
		0.3	214.00	214(g)	0.008	0.002	205.46	2277	0.077	0.001	356.97	2261	0.347	0.001
		0.4	199.00	199(g)	0.009	0.002	192.00	2309	0.125	0.001	282.97	2294	0.337	0.001
	100	0.2	368.00	368(g)	0.033	0.009	416.66	4707	1.244	0.001	557.04	4730	2.087	0.001
		0.3	233.00	233(g)	0.046	0.013	273.74	4770	1.061	0.002	401.15	4812	4.939	0.002
		0.4	192.00	192(g)	0.062	0.017	195.95	4847	0.828	0.002	292.99	4865	4.901	0.002
	150	0.2	368.00	368(g)	0.100	0.028	465.99	7170	6.618	0.003	606.24	7224	12.996	0.003
		0.3	278.00	278(g)	0.147	0.039	326.04	7319	7.959	0.004	401.01	7292	15.161	0.004
		0.4	239.00	239(g)	0.195	0.054	227.66	7368	5.175	0.005	304.84	7348	15.940	0.005
	200	0.2	413.00	413(g)	0.231	0.063	427.48	9687	10.346	0.004	555.81	9753	29.726	0.004
		0.3	333.00	333(g)	0.336	0.092	315.76	9822	14.801	0.006	448.56	9853	34.584	0.006
		0.4	271.00	271(g)	0.478	0.124	268.53	9905	19.898	0.008	349.81	9886	46.916	0.008
	250	0.2	434.00	434(g)	0.441	0.122	458.96	12,259	32.850	0.007	642.11	12,264	53.922	0.007
		0.3	368.00	368(g)	0.675	0.182	374.26	12,349	38.726	0.010	463.27	12,364	85.059	0.010
		0.4	301.00	301(g)	0.860	0.242	291.94	12,402	21.403	0.013	377.78	12,403	81.280	0.013
	300	0.2	439.00	439(g)	0.755	0.21	471.04	14,792	39.780	0.010	583.20	14,811	91.519	0.010
		0.3	372.00	372(g)	1.118	0.307	390.80	14,866	56.137	0.015	449.08	14,892	111.569	0.029
		0.4	340.00	340(g)	1.544	0.416	332.10	14,916	37.602	0.037	393.64	14,919	143.804	0.021
		Avg	312.83	312.83	0.39	0.11	335.57	8554.61	16.37	0.01	443.79	8565.33	40.85	0.01
		#Opt	18				0				0
20	50	0.2	322.00	322(g)	0.008	0.003	412.03	2167	0.238	0.001	685.04	2124	0.623	0.001
		0.3	200.00	200(g)	0.011	0.004	335.36	2294	0.394	0.001	508.90	2254	0.857	0.001
		0.4	145.00	145(g)	0.014	0.005	205.95	2332	0.224	0.001	343.17	2352	1.135	0.001
	100	0.2	327.00	332(g)	0.076	0.017	369.27	4679	1.835	0.001	632.33	4716	4.369	0.001
		0.3	219.00	219(g)	0.070	0.024	339.09	4804	3.875	0.016	435.21	4789	6.340	0.002
		0.4	192.00	194(g)	0.123	0.031	223.88	4851	3.732	0.002	333.04	4867	8.595	0.002
	150	0.2	405.00	405(g)	0.148	0.051	426.92	7218	8.140	0.003	606.58	7176	15.154	0.003
		0.3	268.00	268(g)	0.224	0.078	325.41	7323	11.363	0.004	440.81	7343	19.361	0.004
		0.4	214.00	214(g)	0.301	0.104	261.55	7367	10.763	0.005	405.78	7342	25.113	0.005
	200	0.2	411.00	411(g)	0.354	0.124	483.75	9753	21.034	0.005	610.67	9786	29.232	0.005
		0.3	302.00	302(g)	0.524	0.184	360.53	9825	34.685	0.007	461.50	9852	43.715	0.006
		0.4	266.00	266(g)	0.705	0.245	281.10	9880	36.001	0.008	357.64	9893	59.799	0.009
	250	0.2	408.00	408(g)	0.670	0.241	491.46	12,230	44.024	0.007	615.93	12,274	64.576	0.007
		0.3	333.00	333(g)	1.039	0.353	368.08	12,361	51.527	0.010	440.01	12,349	86.206	0.010
		0.4	286.00	286(g)	1.326	0.471	311.09	12,399	62.237	0.029	381.98	12,408	105.304	0.013
	300	0.2	441.00	441(g)	1.163	0.409	490.17	14,783	56.597	0.010	625.83	14,780	97.131	0.010
		0.3	351.00	351(g)	1.721	0.612	402.47	14,890	84.927	0.015	478.84	14,871	141.432	0.029
		0.4	321.00	321(g)	2.291	0.826	321.13	14,913	2.203	0.038	389.70	14,920	145.153	0.037
		Avg	300.61	301.00	0.60	0.21	356.07	8559.39	24.10	0.01	486.28	8560.89	47.45	0.01
		#Opt	16				0				0

reports the results obtained by the Subgradient algorithm for the BLMST problem on the same instances, described in Section 8.1, used to test the mathematical models. As for Table 1, Table 2 and Table 3, the first columns (|L|, n, and d) report the characteristics of the instances. The remaining columns are divided into the three previously described scenarios, reporting for each of them the following columns: LB, UB, Time, and UB time. LB and UB represent the lower and upper bounds computed by the algorithm, respectively. A match between the LB and UB indicates that the solution associated with the identified upper bound is optimal for that instance. Furthermore, when the upper bound value is associated with the solution obtained by running the greedy algorithm (Algorithm 1), (g) appears close to the UB value. Finally, the Time column denotes the total computational time required by the Subgradient algorithm, while UB time refers exclusively to the time taken to compute the upper bound.

In Scenario 1, associated with the least restrictive budget ($\lceil \left|V\right|/2\rceil$), the algorithm performed exceptionally well, finding optimal solutions in 52 out of 54 instances. This scenario also exhibited the smallest average computational time, indicating a lower complexity in solving these instances with respect to the ones of Scenarios 2 and 3, which require increasingly larger computational times. In particular, for $\left|L\right|=5$, the Subgradient algorithm required an average computational time of 0.28, 1.29, and 6.45 s to compute bounds for an instance of Scenarios 1, 2, and 3, respectively; for $\left|L\right|=10$, the same average computational times are 0.39, 16.37, and 40.85, while for $\left|L\right|=20$, they reach 0.60, 24.10, and 47.45 s. In general, a relevant increase in the run time is recorded as the imposed budget becomes tighter, while a gentler increase is observable when the number of labels increases. This trend is consistent with the increase in the average computational times required by the mathematical models to solve an instance, as observed in Section 8.2.1.

On the other side, the Subgradient algorithm did not identify any optimal solution among the instances of Scenarios 2 and 3, highlighting the increasing challenge posed by tighter budget constraints. However, it consistently provides robust lower bounds, demonstrating its efficacy in approximating solutions for the BLMST problem under varying conditions. As the budget constraint becomes tighter, the BLMST problem becomes more complex, affecting the effectiveness of the algorithm. The subgradient approach, despite these challenges, demonstrates a fast and effective method for finding upper and lower bounds, especially in scenarios with less restrictive budgets.

9. Conclusions

In this study, we introduced the BLMST problem, a new variant of the minimum spanning tree problem. We formally proved the NP-hardness of the BLMST problem and we generated a set of computational instances to evaluate its complexity and practical implications. We developed and compared three distinct mathematical programming formulations: a single commodity flow formulation, a packing formulation, and a penalization-based formulation. These models were tested under several scenarios, characterized by varying budget restrictions, network sizes, and the number of labels. Relevant trends in the performances of the three formulations have been analyzed via a computational test phase, carried out on 162 randomly created instances. The single commodity flow formulation emerged as generally superior in all the scenarios, consistently achieving optimal solutions with competitive computational times. Its performance, however, diminished as the number of labels increased ($\left|L\right|=20$), where it struggled to identify feasible solutions within the time limit. Furthermore, we introduced a Lagrangian approach based on a Subgradient algorithm to derive tighter lower bounds for the optimal solution of the problem, incorporating a greedy strategy to eventually derive upper bounds. Future research might focus on enhancing these formulations or developing hybrid approaches to improve solution quality and computational efficiency, particularly for high-label and tight-budget scenarios. The exploration of heuristic and metaheuristic methods could also provide valuable insights for solving large-scale instances of the BLMST problem.