On Solving the Minimum Spanning Tree Problem with Conflicting Edge Pairs

Abstract

The Minimum Spanning Tree with Conflicting Edge Pairs is a generalization that adds conflict constraints to a classical optimization problem on graphs used to model several real-world applications. In recent years, several heuristic and exact approaches have been proposed to tackle this problem. In this paper, we present a mixed-integer linear program not previously applied to this problem, and we solve it with an open-source solver. Computational results for the benchmark instances commonly adopted in the literature of the problem are reported. The results indicate that the approach we propose obtains results aligned with those of the much more sophisticated approaches available, notwithstanding it being much simpler to implement. During the experimental campaign, six instances were closed for the first time, with nine improved best-known lower bounds and sixteen improved best-known upper bounds over a total of two hundred thirty instances considered.

1. Introduction

The Minimum Spanning Tree Problem with Conflicting Edge Pairs (MSTC) is a variant of the classical minimum spanning tree (MST) problem recently introduced [1]. Given a connected undirected graph with costs on the edges and a set of edge pairs in conflict with each other, the MSTC consists of finding the spanning tree with the minimum total cost that uses at most one of the edges of each conflicting pair. The MSTC was first introduced by Darmann et al. [2]. It was shown that under general settings, the problem cannot be solved in polynomial time, although some special cases are shown to be polynomially solvable. Several heuristic methods were subsequently proposed in [3], while a branch-and-cut based on a mathematical programming model and some valid inequalities was presented in [4]. More recently, a metaheuristic method was designed and presented in [5], while some new methods based on Lagrangian relaxations, able to produce high-quality lower bounds and heuristic solutions, were introduced in [6]. Finally, a new branch-and-cut approach based on some new valid inequalities was discussed in [7] and shown to represent the current state-of-the-art exact solving methods.

In the literature of graph theory and algorithms, several classic optimization problems have been studied in one or more variants involving conflict constraints, witnessing increasing popularity of the topic due to the number of related real-world applications. For example, for the Assignment Problem with Conflicts, theoretical results on the computational complexity, together with some approximation results, can be found in [2]. Further results for special polynomially solvable settings were presented in [8], together with some heuristics and lower-bounding schema. More contributions can be found in [9,10], where some Mixed-Integer Linear Programming models, exact Branch-and-Bound methods, and heuristic ideas are introduced. The concepts of the last two papers were later summarized and extended in [11]. For the Knapsack Problem with Conflicts, a local-search heuristic and a branch-and-bound method based on Lagrangian relaxation are discussed in [12]. In [13], the authors proposed some preprocessing techniques and a different branch-and-bound algorithm. Heuristic methods for this problem were discussed in [14,15]. Metaheuristic approaches were finally presented in [16,17]. Finally, a branch-and-bound exact method, based on binary branching, was proposed in [18], while another branch-and-bound schema, based on n-ary branching, was developed and tested in [19]. Different Set-Covering Problems with Conflicts were proposed in [20,21], together with approximation algorithms and computational complexity results. Different families of valid inequalities for a model representing the problem were presented in [22], together with some preprocessing and speed-up routines. A specialization of the same problem in geometric settings was discussed in [23], where some approximation results were presented. The Shortest Path Problem with Conflicts can be found in [24], where critical variants of the shortest path problem arising in the automatic generation of test paths for programs were introduced, and in [25], where the first definition of conflict pairs was presented, although nodes instead of arcs were a part of the conflicts. A procedure to solve the problem was discussed in [26]. A polyhedral study of the problem was presented in [27], while branch-and-bound and dynamic programming approaches were recently proposed in [28,29]. The Maximum Flow Problem with Conflicts was first studied in [30], where the NP-hardness of the problem is shown. Later, some mixed-integer linear programming formulations and exact methods, such as a Benders Decomposition, a Branch-and-Bound, and a Russian Doll Search were presented in [31]. More recently, the authors of [32] proposed a heuristic approach for the problem called Kernousel, obtained by merging two metaheuristic approaches, one based on Carousel Greedy [33] and the other on Kernel Search [34].

The MSTC can be used to represent real applications arising in different domains. For example, the design of an offshore wind farm network, where the wind turbines have to be connected together, and the different possible point-to-point connections are characterized by different costs. Technical reasons prevent the use of overlapped cables in such a context [35], and this generates conflict constraints. Other practical applications are in the design and installation of long-distance pipelines, where conflicts typically represent the impossibility of using infrastructures owned by competitor service providers or, alternatively, the impossibility of crossing countries with reciprocal political issues with the same line [36].

In this paper, a compact mixed-integer linear programming model for the MSTC is introduced in the context of the MSTC. It is based on multi-commodity flow concepts and extends known formulations for the MST problem ([37,38]) to the case with conflicts. Notice that multi-commodity flow models are common for problems related to the MST problem (see, for example, [39] for the Min-degree Constrained MST and [40] for the Diameter-Constrained MST). The model we propose is solved by the open-source solver CP-SAT, which is a part of the Google OR-Tools [41] optimization suite. Normally, models based on multi-commodity flow are not very effective in the context of integer programming, due to the poor linear relaxation, but we believe such a formulation might perform well when solved with CP-SAT. Successful application of this solver on optimization problems with characteristics similar to the problem under investigation, such as the Parallel Drone-Scheduling Traveling Salesman Problem and Vehicle Routing Problems [42,43], the Maximum Flow Problem with Conflict Constraints [44], and the Home Healthcare-Routing and Scheduling Problem [45], motivated this study. A comprehensive experimental campaign is conducted and reported on the benchmark instances previously used in the literature.

The overall organization of the paper can be summarized as follows: Section 2 is dedicated to the formal introduction and definition of the Minimum Spanning Tree Problem with Conflicting Edge Pairs. Section 3 presents a compact mixed-integer linear programming model to represent the problem. In Section 4, the new model is validated through an extensive experimental perspective. After the description of the benchmark instances considered and of the methods used for comparison purposes, the detailed results obtained by solving the new model are reported. Some conclusions about the work presented and the results achieved are drawn in Section 5 to close the paper.

2. Problem Description

The Minimum Spanning Tree Problem with Additional Conflicts can be defined on an undirected graph ($G=(V,E)$), with V denoting the set of nodes and E the set of edges. Each edge ($\{i,j\}\in E$) is associated with a cost ($u_{ij}\in {\mathbb{N}}^{+}$). The following feature differentiates the problem from the classic Minimum Spanning Tree Problem and increases the theoretical computational complexity of the problem to $\mathcal{NP}$–hard [2]: Two edges ($\{i,j\},\{k,l\}\in E$) are said to be in conflict if, at most, one of them can be included in a feasible solution. The set of edges in conflict with the edge ($\{i,j\}\in E$) is contained in the set ${\delta }_{ij}$. The goal of the MSTC is to find the minimum-cost-spanning tree that satisfies all the conflict constraints.

An example of an MFPC instance and the optimal solution for the same instance are provided in Figure 1.

3. A Mixed-Integer Linear Programming Model

In this section, a model for the MSTC—obtained by extending a classical model for the Minimum Spanning Tree Problem [37]—is proposed. The compact model is based on multi-commodity flow, characterized by the number of variables in the order of O(${\left|V\right|}^2$) and the number of constraints in the order of O(${\left|V\right|}^2$), making it compact and suitable to be attacked directly by a solver. Conversely, the approaches previously considered in the literature rely on models with an exponential number of constraints [4,7], which require a more complex implementation of dynamic and iterative mechanisms to generate violated constraints during the solving process. Our approach is expected to have a weaker linear relaxation than the exponential models, but we believe it has potential when approached with solvers able to do logical inferences, as is done herein. The new model uses a set (A) of (directed) arcs, obtained by including the two arcs ($(i,j)$ and ($j,i)$) for each edge ($\{i,j\}\in E$). A node (s) is arbitrarily selected as the root (node 0 in our implementation), and the model aims to send one unit of flow from the source to each other node.

The new model can be formally defined as follows: A variable ($x_{ij}\in {\mathbb{Z}}_0^{+}$) is introduced for each $(i,j)\in A$ and takes the value of the flow transiting the arc ($(i,j)\in A$). An additional set of variables is introduced to link the multi-commodity flow formulation to the MSTC. Then, a binary variable ($y_{ij}$) is defined for each edge ($\{i,j\}\in E$). It takes a value of 1 if the edge is a part of the solution and 0 otherwise. The resulting compact model, characterized by a polynomial number of variables and constraints, is as follows:

$$\begin{array}{ccc}\hfill & min\sum _{\{i,j\}\in E}{u}_{ij}{y}_{ij}\hfill & \hfill \end{array}$$

(1)

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

(2)

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

(3)

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

(4)

$$\begin{array}{ccc}\hfill & y_{ij}+y_{kl}\le 1\hfill & \hfill \{i,j\}\in E,(k,l)\in {\delta }_{ij}\end{array}$$

(5)

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

(6)

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

(7)

The objective function (1) minimizes the cost of the spanning tree. Constraints (2) implement a classical multi-commodity flow model on the directed network and impose that one unit of the commodity has to exit the source (s) for each other node of V (case $i=s$) and that each node different from s has to retain exactly one unit of the commodity (case $i\ne s$). An example of the flow associated with the solution reported in Figure 1 is available in Figure 2.

Inequalities (3) and (4) connect the x and y variables by imposing that $y_{ij}$ has to take a value of 1 as soon as either $x_{ij}$ or $x_{ji}$ takes a positive value. Constraints (5) encode conflict constraints by ensuring that conflicting edges are not selected simultaneously. Finally, the domains of the variables are specified in constraints (6) and (7) for the x and y variables, respectively.

4. Computational Experiments

This section is devoted to the experimental validation of the new compact model introduced in Section 3. The model’s results are compared to those of existing methods for the standard benchmark instances from the literature. In detail, Section 4.1 is devoted to the description of the benchmark instances previously introduced in the literature and used in the present study. An extensive comparison between the results achieved in this work and the results previously published is presented in Section 4.2.

4.1. Benchmark Instances

The benchmark sets originally proposed in [3,7], which have been widely used in the previous literature to validate and compare solving methods, are adopted for the experiments reported in the present work.

From the benchmark set proposed in [3], the 50 most challenging instances are considered, since the other ones can easily be solved to optimality by any recent method. The instances are generated at random and have a number of nodes that ranges between 50 and 300 and a number of edges between 200 and 1000. The edge costs are between 0 and 500, and different densities for the conflict graph are considered. We refer the interested reader to [3] for full details on instance generation.

A newer set of benchmark instances was recently introduced in [7] and is formed by random instances with the number of nodes between 25 and 100 and the edge density equal to 0.2, 0.3, or 0.4 (notice that the previous set of benchmarks was characterized by lower densities). Costs are between 1 and 30, and the density of the conflict graph is 0.01, 0.04, or 0.07. All 180 instances of the set are considered for the experiments reported in the present work. We refer the reader interested in the generation process of the instances to [7] for full details.

4.2. Experimental Results

The model discussed in Section 3 has been solved with the Google OR-Tools CP-SAT solver [41] version 9.12. The CP-SAT solver is designed to work in a multi-thread environment (compatible with all new processors) and implements a portfolio strategy with effective data exchange among the different threads. The main process runs a Constraint-Programming Solver based on Lazy Clause Generation (LCG) [46]. The concept behind LCG involves the (incremental) transformation of the problem to an SAT formula, subsequently employing an SAT solver to seek a solution (or prove bounds for infeasibility). The input model is also linearized, the corresponding linear program is (partially) solved with the (dual) simplex algorithm, and other classical MILP techniques are run to enhance bounds and retrieve new solutions, aiming at supporting the satisfiability model. Finally, instances of metaheuristic methods, seeking high-quality feasible solutions, are executed.

We believe a compact model heavily based on binary variables, like those discussed in Section 3, might be very suitable to be attacked with a solver with the characteristics of CP-SAT.

The experiments have been run on a computer equipped with an Intel Core i7 12700F CPU, but the computational times were normalized using the Intel Core i7-3770 CPU used for the experiments reported in [5,6,7], as a baseline. This ensures fair comparisons across methods. The normalization of the computation times has been carried out according to the conversion ratios inferred from [47]. Notice that ref. [47] is commonly used in the community to compare results obtained on different machines (see, for example, [32,44]). In accordance with the tests previously reported in [7], a maximum computation time of 5010 s is allowed for all the methods for which a computation time is reported in the tables that follow.

The first experiments have been carried out on the instances proposed in [3] and are summarized in

Table 1. Computational results for the instances introduced in [3].

Instances			MILP	LB-MST	LB-MI	HDA		Mega		HDA+		BC		BC			CP-SAT
			[3]	[3]	[3]	[6]		[5]		[6]		[4]		[7]						Dev%
n	m	p	UB	LB	LB	LB	Sec	UB	Sec	UB	Sec	LB	UB	LB	UB	Sec	LB	UB	Sec	LB	UB
50	200	199	708	701.1	702.8	705.5	0.3	708	0.7	726	0.4	708.0	708	708.0	708	0.2	708	708	5.1	0.0	0.0
50	200	398	770	739.8	757.8	761.0	0.3	770	0.7	780	0.5	770.0	770	770.0	770	0.5	770	770	9.2	0.0	0.0
50	200	597	917	782.7	807.8	867.7	0.4	917	0.6	1176	0.6	917.0	917	917.0	917	1.8	917	917	46.0	0.0	0.0
50	200	995	1324	835.7	877.5	961.9	0.4	1365.4	0.6	1587	0.8	1324.0	1324	1324.0	1324	7.4	1324	1324	199.6	0.0	0.0
50	200	3903	1636	775.1	877.5	1042.8	1.2	1636	0.5	1653	2.2	-	-	-	-	-	1636	1636	0.1	0.0	0.0
50	200	4877	2043	698.7	887.5	1116.3	1.6	2043	0.5	2043	2.5	-	-	-	-	-	2043	2043	0.2	0.0	0.0
50	200	5864	2338	626.9	1030.3	2338.0	0.8	2338	0.6	2338	0.8	-	-	-	-	-	2338	2338	0.1	0.0	0.0
100	300	448	4041	3893.5	3991.2	4036.6	0.9	4099.2	0.6	4099	1.2	4041.0	4041	4041.0	4041	4.6	4041	4041	232.0	0.0	0.0
100	300	897	5658	4508.2	4624.2	4982.0	1.4	-	1.9	-	2.1	5658.0	5658	5658.0	5658	178.5	5658	5658	2758.6	0.0	0.0
100	300	1344	-	-	4681.3	-	-	-	2.9	-	-	6621.2	-	6635.4	-	5010.0	Infeas	Infeas	104.6	−100.0	−100.0
100	300	8609	7434	4043.4	5754.9	7434.0	0.6	7434	1.9	7434	0.6	-	-	-	-	-	7434	7434	0.1	0.0	0.0
100	300	10,686	7968	3970.5	6192.3	7968.0	0.6	7968	1.8	7968	0.5	-	-	-	-	-	7968	7968	0.2	0.0	0.0
100	300	12,761	8166	3936.3	6758.6	8166.0	0.5	8166	1.9	8166	0.5	-	-	-	-	-	8166	8166	0.1	0.0	0.0
100	500	1247	4275	4124.5	4165.7	4268.6	2.3	4291.2	5.3	4301	3.0	4275.0	4275	4275.0	4275	11.5	4275	4275	86.6	0.0	0.0
100	500	2495	5997	4701.9	4805.4	5238.2	2.5	6325	5.0	8214	5.5	5951.4	6006	5997.0	5997	1239.4	5676	6005	5010.0	5.4	0.1
100	500	3741	7665	4743.8	4871.3	5418.8	2.6	7788	3.2	-	6.8	6510.8	9440	6707.8	8049	5010.1	6267	7787	5010.0	6.6	−0.013
100	500	6237	-	-	4969.0	-	-	-	5.0	-	-	7568.7	-	7729.3	-	5010.0	7259	-	5010.0	6.1	0.0
100	500	12,474	-	-	5194.7	-	-	-	6.9	-	-	9816.9	-	10,560.2	-	5010.0	Infeas	Infeas	15.1	−100.0	−100.0
100	500	24,740	12,652	4289.9	5104.9	5565.2	9.0	12,652	3.4	12,681	20.8	-	-	-	-	-	12,652	12,652	0.1	0.0	0.0
100	500	30,886	11,232	3972.7	5078.8	5672.7	15.5	11,232	3.6	11,287	26.4	-	-	-	-	-	11,232	11,232	0.1	0.0	0.0
100	500	36,827	11,481	3918.1	5710.8	11,481.0	22.9	11,481	3.6	11,481	24.4	-	-	-	-	-	11,481	11,481	0.2	0.0	0.0
200	400	13,660	17,728	14,085.8	17,245.9	17,728.0	1.3	17,728	7.7	17,728	1.4	-	-	-	-	-	17,728	17,728	0.1	0.0	0.0
200	400	17,089	18,617	14,067.3	18,048.2	18,617.0	1.5	18,617	7.7	18,617	1.4	-	-	-	-	-	18,617	18,617	0.1	0.0	0.0
200	400	20,470	19,140	13,998.7	18,646.2	19,140.0	1.4	19,140	7.6	19,140	1.4	-	-	-	-	-	19,140	19,140	0.1	0.0	0.0
200	600	1797	15,029	1111.6	11,425.8	12,451.6	4.9	-	12.6	-	8.6	13,072.9	14,707	13,171.2	14,086	5010.0	12,841	14,405	5010.0	2.5	2.3
200	600	3594		-	12,487.0		-	-	24.0	-	-	17,532.7	-	17,595.0	-	5010.0	0	0	158.9	−100.0	−100.0
200	600	5391		-	12,873.2		-	-	31.6	-	-	Infeas	Infeas	Infeas	Infeas	16.4	Infeas	Infeas	0.0	0.0	0.0
200	600	34,504	20,716	9466.1	15,393.1	20,716.0	3.0	20,716	11.6	20,716	3.0	-	-	-	-	-	20,716	20,716	0.1	0.0	0.0
200	600	42,860	18,025	9100.6	13,971.5	18,025.0	2.1	18,025	12.0	18,025	2.2	-	-	-	-	-	18,025	18,025	0.2	0.0	0.0
200	600	50,984	20,864	8734.5	16,708.1	20,864.0	1.9	20,864	12.5	20,864	2.0	-	-	-	-	-	20,864	20,864	0.1	0.0	0.0
200	800	3196	22,110	17,428.8	17,922.6	19,685.1	8.6	22,350.8	22.2	-	18.9	20,744.2	21,852	20,941.5	21,553	5010.1	20,030	22,722	5010.0	4.4	5.4
200	800	6392	-	-	19,705.7	-		-	28.4	-	-	26,361.3	-	26,526.7	-	5010.1	24,259	-	5010.0	8.5	0.0
200	800	9588	-	-	20,684.8	-		-	38.3	-	-	29,443.6	-	30,634.2	-	5010.0	28,351	-	5010.0	7.5	0.0
200	800	15,980	-	-	20,226.9	-		-	48.0	-	-	33,345.1	-	36,900.2	-	5010.0	0	0	35.4	−100.0	−100.0
200	800	62,625	39,895	16,806.5	23,792.3	39,895.0	43.3	39,895	16.7	39,895	45.8	-	-	-	-	-	39,895	39,895	0.2	0.0	0.0
200	800	78,387	37,671	15,803.1	22,174.2	37,671.0	7.7	37,671	15.9	37,671	8.2	-	-	-	-	-	37,671	37,671	0.2	0.0	0.0
200	800	93,978	38,798	15,470.1	24,907.0	38,798.0	4.5	38,798	17.3	38,798	4.7	-	-	-	-	-	38,798	38,798	0.1	0.0	0.0
300	600	31,000	43,721	34,154.2	42,720.6	43,721.0	3.0	43,721	19.2	43,721	3.4	-	-	-	-	-	43,721	43,721	0.1	0.0	0.0
300	600	38,216	44,267	33,320.1	43,486.7	44,267.0	3.1	44,267	22.3	44,267	3.4	-	-	-	-	-	44,267	44,267	0.2	0.0	0.0
300	600	45,310	43,071	32,072.3	42,149.0	43,071.0	3.0	43,071	24.6	43,071	3.1	-	-	-	-	-	43,071	43,071	0.3	0.0	0.0
300	800	3196	-	-	30,190.1	-	-	-	62.8	-	-	Infeas	Infeas	Infeas	Infeas	2911.1	Infeas	Infeas	3.3	0.0	0.0
300	800	59,600	43,125	24,384.3	36,629.6	43,125.0	3.3	43,125	32.7	43,125	3.1	-	-	-	-	-	43,125	43,125	0.1	0.0	0.0
300	800	74,500	42,292	22,913.2	38,069.3	42,292.0	3.2	42,292	35.6	42,292	3.2	-	-	-	-	-	42,292	42,292	0.3	0.0	0.0
300	800	89,300	44,114	21,624.6	38,843.0	44,114.0	3.4	44,114	35.4	44,114	3.2	-	-	-	-	-	44,114	44,114	0.1	0.0	0.0
300	1000	4995	-	-	40,732.7	-	-	-	79.3	-	-	51,451.3	-	51,398.4	-	5010.0	48,180	-	5010.0	6.4	0.0
300	1000	9990	-	-	42,902.5	-	-	-	119.1	-	-	60,907.8	-	61,878.9	-	5010.0	Infeas	Infeas	181.3	−100.0	−100.0
300	1000	14,985	-	-	44,639.1	-	-	-	141.3	-	-	Infeas	Infeas	Infeas	Infeas	1820.0	Infeas	Infeas	9.7	0.0	0.0
300	1000	96,590	71,562	36,544.7	56,048.3	71,562.0	9.2	71,562	41.5	71,562	9.5	-	-	-	-	-	71,562	71,562	0.1	0.0	0.0
300	1000	120,500	76,345	34,380.8	58,780.1	76,345.0	6.6	76,345	33.4	76,345	6.7	-	-	-	-	-	76,345	76,345	0.2	0.0	0.0
300	1000	144,090	78,880	33,481.2	60,810.8	78,880.0	5.2	78,880	36.9	78,880	5.4	-	-	-	-	-	78,880	78,880	0.1	0.0	0.0
Averages																				−9.1	−9.8

. The first columns identify each instance (n is the number of nodes, m is the number of edges, and p is the number of conflicts, following the parameters used to generate the instances), while the rest of the table summarizes all the relevant results presented in the previous literature. The (not necessarily) optimal results obtained by solving a Mixed-Integer Linear Programming (MILP) model and two lower-bounding procedures (LB-MST and LB-MI) are taken from [3] (with no computation time). The lower and upper bounds (HDA and HDA+, respectively) from [6] are reported together with the upper bounds obtained with the multi-ethnic genetic metaheuristic method that appeared in [5] (Mega) and the lower and upper bounds obtained by the two branch-and-cut approaches discussed in [4] (unfortunately, without computation times) and [7], respectively (BC). Finally, the results obtained by the method presented in this paper are presented in the last five columns (CP-SAT). On top of the lower and upper bounds obtained and the relative computation time, the percentage deviation (Dev%) against the best lower and upper bounds is also reported to better position the new approach. These deviations are calculated as $100·{\displaystyle \frac{B{K}_{LB}-LB}{B{K}_{LB}}}$ and $100·{\displaystyle \frac{UB-B{K}_{UB}}{B{K}_{UB}}}$, respectively, where $B{K}_{LB}$ and $B{K}_{UB}$ are the best-known lower and upper bounds available from the previous literature, and $LB$ and $UB$ are the values provided by CP-SAT. Negative deviations (corresponding to improved best-known results) are highlighted in bold.

The results in Table 1 suggest that the approach we propose (CP-SAT) is particularly effective in identifying infeasible instances: Five such instances were identified for the first time in this study by the powerful feasibility checker provided by the adopted solver (notice that infeasible instances are marked as Infeas in the table). This corroborates the intuition that the logical inference carried out by CP-SAT can be effective on the compact model we propose. Moreover, another best-known upper bound has been improved. In general, the new approach has a performance comparable with those of the other state-of-the-art methods available, with deviation gaps remaining below 8.5% (and, in general, substantially below) from best-known bounds (taken as the best of the results obtained by several different algorithms). We observe that the average deviations reported are not very significant because they are heavily affected by the figures for the infeasibility proofs. Notably, CP-SAT consistently provides meaningful lower and upper bounds, demonstrating a level of robustness—especially on the upper bounds—that is uncommon among other methods. On the other hand, the lower bounds found by CP-SAT appear sometimes worse than those of BC (from [7]) in some of the most challenging instances. This might indicate scalability issues. When observing computation times, CP-SAT often appears slower than the other methods. This likely depends on the model we propose, which is characterized by a poor linear relaxation, which, in turn, means that no strong lower bound from linear programming is available to the solver. CP-SAT effectively compensates for this lack of information by applying logical inference, although this takes a longer time, showing that the solver is suitable for the new model, and the combination leads to very good results.

The percentage deviations from the best-known (before the present study) lower and upper bounds are plotted for all the methods considered and for each instance of the first benchmark set in Figure 3 and Figure 4. The plots help to position the new approach even more clearly. A plot representing the computation times required to close each instance by the two exact methods for which such information is available (BC [7] and CP-SAT) is provided in Figure 5. A negative value is plotted when no result is available for the solver BC. The plot suggests how CP-SAT is often generally slower in the instances solved by the two methods, but, on the other hand, the number of instances solved by such an approach is higher.

A second set of experiments has been carried out on the instances proposed in [7], and the results are summarized in

Table 2. Computational results for the instances introduced in [7].

Instances				HDA		HDA+		BC			CP-SAT
				[6]		[6]		[7]						Dev%
$\mathit{n}$	$\mathit{m}$	$\mathit{p}$	$\mathit{s}$	LB	Sec	UB	Sec	LB	UB	Sec	LB	UB	Sec	LB	UB
25	60	18	1	347.0	0.04	347	0.05	347.0	347	0.0	347	347	0.23	0.0	0.0
25	60	18	7	389.0	0.04	389	0.04	389.0	389	0.0	389	389	0.18	0.0	0.0
25	60	18	13	353.0	0.04	353	0.04	353.0	353	0.0	353	353	0.19	0.0	0.0
25	60	18	19	346.0	0.04	346	0.05	346.0	346	0.0	346	346	0.19	0.0	0.0
25	60	18	25	336.0	0.04	336	0.04	336.0	336	0.0	336	336	0.18	0.0	0.0
25	90	71	31	379.6	0.09	381	0.07	381.0	381	0.0	381	381	0.45	0.0	0.0
25	60	71	37	381.5	0.05	390	0.07	390.0	390	0.1	390	390	0.32	0.0	0.0
25	60	71	43	372.0	0.05	372	0.06	372.0	372	0.0	372	372	0.44	0.0	0.0
25	60	71	49	357.0	0.05	357	0.06	357.0	357	0.0	357	357	0.24	0.0	0.0
25	60	71	55	406.0	0.05	406	0.06	406.0	406	0.0	406	406	0.38	0.0	0.0
25	60	124	61	385.0	0.06	385	0.07	385.0	385	0.0	385	385	0.18	0.0	0.0
25	60	124	67	432.0	0.06	432	0.07	432.0	432	0.0	432	432	0.20	0.0	0.0
25	60	124	73	424.5	0.07	474	0.1	458.0	458	0.3	458	458	0.48	0.0	0.0
25	60	124	79	398.0	0.07	400	0.08	400.0	400	0.0	400	400	0.45	0.0	0.0
25	60	124	85	406.8	0.08	421	0.12	420.0	420	0.0	420	420	0.42	0.0	0.0
25	90	41	91	310.1	0.05	311	0.05	311.0	311	0.0	311	311	0.54	0.0	0.0
25	90	41	97	306.0	0.05	306	0.06	306.0	306	0.0	306	306	0.36	0.0	0.0
25	90	41	103	299.0	0.05	299	0.05	299.0	299	0.0	299	299	0.31	0.0	0.0
25	90	41	109	297.0	0.05	297	0.05	297.0	297	0.0	297	297	0.30	0.0	0.0
25	90	41	115	318.0	0.05	318	0.05	318.0	318	0.0	318	318	0.25	0.0	0.0
25	90	161	121	305.0	0.08	305	0.09	305.0	305	0.0	305	305	0.54	0.0	0.0
25	90	161	127	339.0	0.12	339	0.11	339.0	339	0.0	339	339	0.29	0.0	0.0
25	90	161	133	344.0	0.08	344	0.09	344.0	344	0.0	344	344	0.23	0.0	0.0
25	90	161	139	328.0	0.09	331	0.11	329.0	329	0.0	329	329	0.61	0.0	0.0
25	90	161	145	325.0	0.08	327	0.11	326.0	326	0.0	326	326	0.89	0.0	0.0
25	90	281	151	347.9	0.1	349	0.13	349.0	349	0.0	349	349	0.56	0.0	0.0
25	90	281	157	370.6	0.1	385	0.16	385.0	385	0.5	385	385	2.67	0.0	0.0
25	90	281	163	330.8	0.11	335	0.14	335.0	335	0.0	335	335	0.66	0.0	0.0
25	90	281	169	334.7	0.1	358	0.18	348.0	348	0.1	348	348	1.91	0.0	0.0
25	90	281	175	350.1	0.1	359	0.17	357.0	357	0.0	357	357	0.98	0.0	0.0
25	120	72	181	282.0	0.07	282	0.06	282.0	282	0.0	282	282	0.24	0.0	0.0
25	120	72	187	294.0	0.06	294	0.06	294.0	294	0.0	294	294	0.90	0.0	0.0
25	120	72	193	284.0	0.06	284	0.08	284.0	284	0.0	284	284	0.48	0.0	0.0
25	120	72	199	281.0	0.06	281	0.07	281.0	281	0.0	281	281	0.53	0.0	0.0
25	120	72	205	292.0	0.06	292	0.06	292.0	292	0.0	292	292	0.30	0.0	0.0
25	120	286	211	320.3	0.12	321	0.16	321.0	321	0.0	321	321	1.06	0.0	0.0
25	120	286	217	317.0	0.12	317	0.17	317.0	317	0.0	317	317	0.64	0.0	0.0
25	120	286	223	284.0	0.56	284	0.12	284.0	284	0.0	284	284	0.31	0.0	0.0
25	120	286	229	311.0	0.12	312	0.17	311.0	311	0.0	311	311	0.58	0.0	0.0
25	120	286	235	290.0	0.13	290	0.13	290.0	290	0.0	290	290	0.36	0.0	0.0
25	120	500	241	319.2	0.16	341	0.26	329.0	329	0.1	329	329	4.72	0.0	0.0
25	120	500	247	326.1	0.16	347	0.25	339.0	339	0.5	339	339	1.84	0.0	0.0
25	120	500	253	354.5	0.16	383	0.3	368.0	368	0.4	368	368	1.55	0.0	0.0
25	120	500	259	306.5	0.16	314	0.22	311.0	311	0.0	311	311	0.65	0.0	0.0
25	120	500	265	318.0	0.16	325	0.25	321.0	321	0.0	321	321	0.74	0.0	0.0
50	245	299	271	619.0	0.34	619	0.37	619.0	619	0.0	619	619	2.75	0.0	0.0
50	245	299	277	604.0	0.36	604	0.38	604.0	604	0.0	604	604	1.49	0.0	0.0
50	245	299	283	634.0	0.34	634	0.35	634.0	634	0.0	634	634	0.72	0.0	0.0
50	245	299	289	615.5	0.34	616	0.4	616.0	616	0.1	616	616	1.49	0.0	0.0
50	245	299	295	595.0	0.34	595	0.44	595.0	595	0.0	595	595	7.28	0.0	0.0
50	245	1196	301	668.5	0.57	698	0.96	678.0	678	1.4	678	678	27.19	0.0	0.0
50	245	1196	307	655.5	0.56	721	0.99	681.0	681	3.2	681	681	51.96	0.0	0.0
50	245	1196	313	678.2	0.59	725	1	709.0	709	6.3	709	709	612.25	0.0	0.0
50	245	1196	319	634.0	1.26	656	0.86	639.0	639	1.5	639	639	16.54	0.0	0.0
50	245	1196	325	658.8	0.56	748	1.01	681.0	681	3.8	681	681	57.54	0.0	0.0
50	245	2093	331	661.3	0.76	-	-	791.2	820	5010.0	778	816	5010.00	1.7	−0.5
50	245	2093	337	706.0	0.75	-	-	835.0	835	1938.7	835	835	4603.22	0.0	0.0
50	245	2093	343	666.5	0.74	-	-	773.2	811	5010.1	760	822	5010.00	1.7	1.4
50	245	2093	349	680.8	0.76	-	-	820.0	836	5010.1	832	832	2874.48	−1.5	−0.5
50	245	2093	355	693.9	0.74	-	-	769.0	769	25.7	769	769	266.66	0.0	0.0
50	367	672	361	570.0	0.82	570	0.88	570.0	570	0.1	570	570	8.05	0.0	0.0
50	367	672	367	561.0	0.86	561	0.98	561.0	561	1.4	561	561	10.48	0.0	0.0
50	367	672	373	573.0	0.86	573	1.06	573.0	573	0.0	573	573	2.98	0.0	0.0
50	367	672	379	560.0	0.86	560	0.87	560.0	560	0.0	560	560	1.61	0.0	0.0
50	367	672	385	549.0	0.83	551	0.99	549.0	549	0.5	549	549	17.12	0.0	0.0
50	367	2687	391	596.4	1.26	657	2.37	612.0	612	7.5	612	612	230.76	0.0	0.0
50	367	2687	397	596.7	1.27	663	2.24	615.0	615	6.6	615	615	154.67	0.0	0.0
50	367	2687	403	577.1	1.26	635	2.33	587.0	587	3.0	587	587	35.79	0.0	0.0
50	367	2687	409	608.5	1.26	721	2.4	634.0	634	7.3	634	634	508.17	0.0	0.0
50	367	2687	415	636.1	3.08	688	2.45	643.0	643	3.2	643	643	72.48	0.0	0.0
50	367	4702	421	624.4	1.66	-	-	701.3	726	5010.1	689	735	5010.00	1.7	1.2
50	367	4702	427	635.4	1.65	-	-	719.5	753	5010.0	712	762	5010.00	1.0	1.2
50	367	4702	433	646.9	1.64	-	-	723.9	786	5010.0	719	784	5010.00	0.7	−0.3
50	367	4702	439	597.5	1.72	-	-	669.8	711	5010.0	666	702	5010.00	0.6	−1.3
50	367	4702	445	665.1	1.71	868	3.36	737.3	764	5010.0	727	775	5010.00	1.4	1.4
50	490	1199	451	548.0	1.68	552	2.03	548.0	548	0.1	548	548	8.41	0.0	0.0
50	490	1199	457	530.0	1.66	531	1.88	530.0	530	0.5	530	530	11.77	0.0	0.0
50	490	1199	463	549.0	1.65	549	1.76	549.0	549	0.0	549	549	3.08	0.0	0.0
50	490	1199	469	540.0	1.67	541	2.01	540.0	540	0.2	540	540	46.21	0.0	0.0
50	490	1199	475	540.0	1.65	540	1.71	540.0	540	0.0	540	540	4.77	0.0	0.0
50	490	4793	481	584.3	2.38	629	4.63	594.0	594	7.8	594	594	109.31	0.0	0.0
50	490	4793	487	559.9	2.4	650	4.9	579.0	579	13.8	579	579	1473.29	0.0	0.0
50	490	4793	493	578.3	2.43	657	4.75	589.0	589	3.0	589	589	31.17	0.0	0.0
50	490	4793	499	565.4	2.39	643	4.74	577.0	577	7.5	577	577	118.57	0.0	0.0
50	490	4793	505	577.4	2.4	670	4.71	592.0	592	6.0	592	592	40.34	0.0	0.0
50	490	8387	511	575.0	2.4	812	6.54	632.2	666	5010.0	626	683	5010.00	1.0	2.6
50	490	8387	517	569.0	3.07	-	-	626.7	651	5010.0	618	646	5010.00	1.4	−0.8
50	490	8387	523	592.2	3.08	-	-	658.4	678	5010.0	650	676	5010.00	1.3	−0.3
50	490	8387	529	592.5	3.07	-	-	662.2	682	5010.1	654	692	5010.00	1.2	1.5
50	490	8387	535	587.5	3.09	828	6.95	644.2	657	5010.0	636	674	5010.00	1.3	2.6
75	555	1538	541	868.0	2.51	-	-	868.0	868	0.7	868	868	75.24	0.0	0.0
75	555	1538	547	870.9	2.49	-	-	871.0	871	3.0	871	871	58.57	0.0	0.0
75	555	1538	553	838.0	2.45	-	-	838.0	838	0.3	838	838	33.73	0.0	0.0
75	555	1538	559	855.0	2.47	-	-	855.0	855	4.4	855	855	44.64	0.0	0.0
75	555	1538	565	857.0	2.47	-	-	857.0	857	4.1	857	857	48.19	0.0	0.0
75	555	6150	571	942.3	3.49	-	-	1023.7	1047	5010.0	1011	1048	5010.00	1.2	0.1
75	555	6150	577	937.6	3.54	-	-	1008.8	1069	5010.2	996	1054	5010.00	1.3	−1.4
75	555	6150	583	910.9	3.48	-	-	987.3	1040	5010.1	973	1044	5010.00	1.4	0.4
75	555	6150	589	913.0	3.53	-	-	985.6	998	5010.1	973	1004	5010.00	1.3	0.6
75	555	6150	595	897.9	3.54	-	-	962.6	994	5010.1	953	1009	5010.00	1.0	1.5
75	555	10,762	601	922.5	4.44	-	-	1054.5	-	4647.3	1048	-	5010.00	0.6	0.0
75	555	10,762	607	948.0	12.01	-	-	1070.9	-	3483.6	1066	-	5010.00	0.5	0.0
75	555	10,762	613	901.9	4.44	-	-	1041.0	-	5010.0	1034	-	5010.00	0.7	0.0
75	555	10,762	619	888.4	4.44	-	-	1006.4	-	5010.0	1005	-	5010.00	0.1	0.0
75	555	10,762	625	914.7	4.42	-	-	1048.1	-	3208.1	1043	-	5010.00	0.5	0.0
75	832	3457	631	797.8	6.86	-	-	798.0	798	4.8	798	798	78.24	0.0	0.0
75	832	3457	637	820.0	6.83	-	-	821.0	821	1.1	821	821	637.56	0.0	0.0
75	832	3457	643	815.3	6.87	-	-	816.0	816	4.0	816	816	262.59	0.0	0.0
75	832	3457	649	820.0	7.16	-	-	820.0	820	23.9	820	820	184.06	0.0	0.0
75	832	3457	655	815.0	7.23	-	-	815.0	815	1.4	815	815	84.81	0.0	0.0
75	832	13,828	661	830.9	9.08	-	-	875.7	891	5010.0	867	909	5010.00	1.0	2.0
75	832	13,828	667	859.5	9.05	-	-	901.8	947	5010.1	893	956	5010.00	1.0	1.0
75	832	13,828	673	829.0	9.04	-	-	873.7	892	5010.0	862	934	5010.00	1.3	4.7
75	832	13,828	679	842.2	8.98	-	-	885.6	915	4570.8	874	918	5010.00	1.3	0.3
75	832	13,828	685	846.7	9.05	-	-	886.9	896	5010.0	877	911	5010.00	1.1	1.7
75	832	24,199	691	870.0	12.24	-	-	955.8	-	1305.3	954	-	5010.00	0.2	0.0
75	832	24,199	697	837.7	12.33	-	-	913.0	-	1161.2	913	-	5010.00	0.0	0.0
75	832	24,199	703	842.0	27.17	-	-	915.0	-	953.0	913	-	5010.00	0.2	0.0
75	832	24,199	709	863.0	11.98	-	-	950.4	-	5010.1	954	-	5010.00	−0.4	0.0
75	832	24,199	715	878.7	12.14	-	-	960.8	-	962.3	960	-	5010.00	0.1	0.0
75	1110	6155	721	787.0	14.88	-	-	787.0	787	2.1	787	787	237.82	0.0	0.0
75	1110	6155	727	785.0	14.82	-	-	785.0	785	5.4	785	785	214.71	0.0	0.0
75	1110	6155	733	783.0	14.78	-	-	783.0	783	0.0	783	783	102.86	0.0	0.0
75	1110	6155	739	783.9	14.94	-	-	784.0	784	3.3	784	784	128.21	0.0	0.0
75	1110	6155	745	796.5	14.86	-	-	797.0	797	5.7	797	797	479.97	0.0	0.0
75	1110	24,620	751	816.9	19.16	-	-	846.5	857	5010.0	840	871	5010.00	0.8	1.6
75	1110	24,620	757	798.0	19.28	-	-	829.2	851	4573.5	824	847	5010.00	0.6	−0.5
75	1110	24,620	763	808.4	19.44	-	-	841.5	892	2189.8	837	917	5010.00	0.5	2.8
75	1110	24,620	769	808.0	19.31	-	-	841.6	864	2866.0	833	876	5010.00	1.0	1.4
75	1110	24,620	775	804.0	19.32	-	-	837.8	868	5010.1	829	868	5010.00	1.0	0.0
75	1110	43,085	781	817.0	27.33	-	-	873.5	-	2357.1	870	-	5010.00	0.4	0.0
75	1110	43,085	787	804.0	26.74	-	-	857.0	-	3106.8	859	-	5010.00	−0.2	0.0
75	1110	43,085	793	827.3	26.9	-	-	885.9	1194	3443.8	884	-	5010.00	0.2	100.0
75	1110	43,085	799	799.0		-	-	856.70	-	3448.3	861	-	5010.00	−0.5	0.0
75	1110	43,085	805	806.0	27.14	-	-	859.1	-	3146.2	851	-	5010.00	0.9	0.0
100	990	4896	811	1118.5	11.59	-	-	1119.0	1119	43.7	1116	1121	5010.00	0.3	0.2
100	990	4896	817	1134.7	11.43	-	-	1137.0	1137	11.8	1137	1137	1091.57	0.0	0.0
100	990	4896	823	1112.5	11.71	-	-	1113.0	1113	71.8	1113	1113	493.89	0.0	0.0
100	990	4896	829	1109.6	11.48	-	-	1110.0	1110	48.6	1110	1110	942.31	0.0	0.0
100	990	4896	835	1088.7	11.99	-	-	1090.0	1090	35.8	1090	1090	357.10	0.0	0.0
100	990	19,583	841	1175.6	15.37	-	-	1249.4	-	5010.0	1239	1570	5010.00	0.8	−100.0
100	990	19,583	847	1142.0	14.98	-	-	1225.8	1491	5010.0	1205	1650	5010.00	1.7	10.7
100	990	19,583	853	1140.7	14.8	-	-	1215.0	1510	5010.0	1203	1623	5010.00	1.0	7.5
100	990	19,583	859	1178.1	14.82	-	-	1264.2	1441	5010.2	1248	1632	5010.00	1.3	13.3
100	990	19,583	865	1176.4	15.36	-	-	1257.3	1560	5010.1	1244	1555	5010.00	1.1	−0.3
100	990	34,269	871	1126.4	22.2	-	-	1264.8	-	5010.0	1251	-	5010.00	1.1	0.0
100	990	34,269	877	1148.9	24.59	-	-	1295.6	-	5010.0	1285	-	5010.00	0.8	0.0
100	990	34,269	883	1179.9	21.17	-	-	1323.7	-	5010.0	1305	-	5010.00	1.4	0.0
100	990	34,269	889	1146.7	20.3	-	-	1283.6	-	5010.0	1268	-	5010.00	1.2	0.0
100	990	34,269	895	1155.0	20.42	-	-	1309.1	-	5010.0	1297	-	5010.00	0.9	0.0
100	1485	11,019	901	1077.9	34.81	-	-	1079.0	1079	248.6	1077	1081	5010.00	0.2	0.2
100	1485	11,019	907	1054.9	34.46	-	-	1056.0	1056	113.4	1056	1056	757.17	0.0	0.0
100	1485	11,019	913	1059.0	34.29	-	-	1059.0	1059	47.9	1059	1059	503.94	0.0	0.0
100	1485	11,019	919	1046.0	34.09	-	-	1046.0	1046	195.2	1046	1046	1206.51	0.0	0.0
100	1485	11,019	925	1071.2	34.17	-	-	1072.0	1072	249.6	1071	1072	5010.00	0.1	0.0
100	1485	44,075	931	1096.9	46.07	-	-	1144.0	1374	3018.3	1130	1444	5010.00	1.2	5.1
100	1485	44,075	937	1091.0	45.6	-	-	1143.6	1291	2144.6	1134	1511	5010.00	0.8	17.0
100	1485	44,075	943	1084.4	46.54	-	-	1137.6	1344	3075.6	1125	1390	5010.00	1.1	3.4
100	1485	44,075	949	1089.4	46.01	-	-	1136.9	1286	3523.3	1131	1417	5010.00	0.5	10.2
100	1485	44,075	955	1088.4	45.84	-	-	1134.6	1370	2954.2	1127	1524	5010.00	0.7	11.2
100	1485	77,131	961	1083.5	60.68	-	-	1164.4	-	4773.8	1156	-	5010.00	0.7	0.0
100	1485	77,131	967	1086.0	60.29	-	-	1168.2	-	5010.0	1160	-	5010.00	0.7	0.0
100	1485	77,131	973	1094.0	60.69	-	-	1187.8	-	5010.0	1175	-	5010.00	1.1	0.0
100	1485	77,131	979	1105.9	60.83	-	-	1183.7	-	5010.0	1175	-	5010.00	0.7	0.0
100	1485	77,131	985	1090.9	60.63	-	-	1164.5	-	5010.0	1155	-	5010.00	0.8	0.0
100	1980	19,593	991	1030.9	76.15	-	-	1031.0	1031	214.4	1031	1031	919.88	0.0	0.0
100	1980	19,593	997	1034.9	76.02	-	-	1036.0	1036	42.8	1036	1036	1298.69	0.0	0.0
100	1980	19,593	1003	1024.0	76.43	-	-	1024.0	1024	21.8	1024	1024	622.64	0.0	0.0
100	1980	19,593	1009	1025.0	76.67	-	-	1025.0	1025	27.4	1025	1025	546.97	0.0	0.0
100	1980	19,593	1015	1027.5	77.49	-	-	1028.0	1028	151.0	1028	1028	1964.48	0.0	0.0
100	1980	78,369	1021	1060.5	106.68	-	-	1097.0	1234	5010.1	1091	1358	5010.00	0.5	10.0
100	1980	78,369	1027	1034.0	106.82	-	-	1065.6	1187	5010.1	1064	1305	5010.00	0.2	9.9
100	1980	78,369	1033	1050.5	104.96	-	-	1087.4	1213	5010.1	1076	1314	5010.00	1.0	8.3
100	1980	78,369	1039	1048.1	103.83	-	-	1083.5	1221	5010.1	1076	1320	5010.00	0.7	8.1
100	1980	78,369	1045	1048.9	103.99	-	-	1084.1	1245	5010.1	1079	1353	5010.00	0.5	8.7
100	1980	137,145	1051	1041.8	129.47	-	-	1100.6	-	5010.1	1086	-	5010.00	1.3	0.0
100	1980	137,145	1057	1066.5	129.26	-	-	1126.3	-	5010.1	1114	-	5010.00	1.1	0.0
100	1980	137,145	1063	1052.4	130.38	-	-	1111.3	-	5010.1	1087	-	5010.00	2.2	0.0
100	1980	137,145	1069	1062.9	130.01	-	-	1118.7	-	5010.1	1100	-	5010.00	1.7	0.0
100	1980	137,145	1075	1055.5	130.01	-	-	1114.1	-	5010.1	1105	-	5010.00	0.8	0.0
Averages														0.4	0.8

. The meaning of the column matches that of the column in Table 1, with the addition of parameter s, which is the seed used to generate the instances and helps to distinguish among instances with the same characteristics. The results of all the methods for which results are available in this dataset (namely, the most recent approaches) are reported. Computational times are available for all the methods reported.

The comparison proposed in Table 2 shows that the approach we propose (CP-SAT) also performs well in this second set of benchmark instances, with average deviations from the previously best-known results of 0.4% concerning the lower bounds and 0.8% concerning the upper bounds. These figures are very good, considering the abundance of methods considered while calculating the (previous) best-known results. The maximum deviation is negligible for most of the instances, with only a few cases above 10% for upper bounds and never above 2.2% for lower bounds. Notice that for some of the instances of this benchmark set, CP-SAT suffers on the heuristic solution side (upper bounds). This indicates that several tailored heuristics presented for the problem were well motivated. A total of four new best-known lower bounds and ten new best-known upper bounds have been produced during the experimental campaign, with one instance (50-245-2093-349) closed for the first time during this experimental campaign. The same conclusion drawn for computation times in Table 1 also applies to these experiments too.

Analogous to what was done for the first benchmark set, the percentage deviations from the best-known (before the present study) lower and upper bounds are plotted for all the methods considered and for each instance of the first benchmark set in Figure 6 and Figure 7. The plots help to position the new approach even more clearly, although the dominance of CP-SAT here is not as clear as in Figure 3 and Figure 4. We plot the computation times required to close each instance by the two exact methods for which such information is available in Figure 8. The plot suggests that for the second benchmark set, BS converges faster (and closes a few more instances) than CP-SAT.

As a general conclusion, the proposed model closely matches or improves upon the best-known results available prior to this work. The proposed method also demonstrates consistent robustness across the entire dataset, although its convergence times appear slower than those of some of the methods previously proposed.

5. Conclusions

A compact mixed-integer linear formulation for the Minimum Spanning Tree Problem with Conflicting Edge Pairs was proposed for the first time in the context of this problem and solved via an open-source solver.

The results of a vast experimental campaign run on the benchmark instances adopted in the previous literature are reported. The results indicate that the proposed approach achieves computational results comparable with the best of those achieved by the substantially more complex methods that appeared in the prior literature, although, sometimes, the computation times are longer due to the mechanics of the solver. A total of six instances were closed for the first time, with nine best-known lower bounds and sixteen best-known upper bounds improved out of the two hundred thirty instances considered. In some of the instances, the new approach still performs slightly worse than some among the other methods it is compared against, but overall, it guarantees robust results, with improvements in instances attacked and solved in the research community for more than a decade.

Future research directions might be twofold. On the one hand, compact models, like the one we propose, might be employed on problems with conflicts similar to the one treated in this paper or employ soft constraints (that can be violated at the price of a penalty), which, by their nature, would pose new challenges. On the other hand, variants of the Minimum Spanning Tree, characterized by soft constraints, with penalties to be paid if violated, might be considered.