THE EDGE-LABELED SURVIVABLE NETWORK DESIGN PROBLEM: FORMULATIONS AND BRANCH-AND-CUT

. In this paper, we study a variant of the survivable network design problem, that is the survivable network design problem with labels (colors) on the edges. In particular, we address the Generalized Labeled Two Edge Connected Subgraph Problem (GLTECSP) that has many applications in telecommunication and transportation. Given a connected undirected graph 𝐺 such that with each edge is associated a set of labels (colors), the GLTECSP consists in finding a two-edge connected spanning subgraph of 𝐺 with a minimum number of distinct labels. We propose two Integer Programming (IP) formulations for the problem, a natural formulation using cuts on the edges, and a compact formulation using color-cuts. We devise Branch-and-Cut algorithms to solve both formulations and compare them on sets of randomly generated instances. Computational results show that the compact formulation outperforms the natural one regarding the linear relaxation and the computational time. Moreover, the compact formulation is able to solve to optimality several instances left unsolved within the time limit by the natural formulation.


Introduction
During recent years, designing survivable telecommunication networks has been of great interest.A telecommunication network is said to be survivable if it remains always connected even in the case of failure of one or many links (or nodes).A telecommunication network can be modeled as an undirected graph  = (, ), where the vertices of  represent telecommunication nodes, and the edges of  the possible links between them.In order to have a minimum degree of survivability of this graph (network), one has to ensure that the graph is 2-edge connected.That is to say between any two vertices of  there exist at least two-edge-disjoint paths.Recall that two paths are said to be edge-disjoint if they do not have any edge in common.Obviously, a 2-edge connected graph remains connected in the case of one-link failure.Given an undirected weighted graph  = (, , ) where (.) is the weight vector on the edges, the Two Edge Connected Subgraph Problem (TECSP) consists in finding a minimum-cost 2-edge connected subgraph of .In this paper, we are interested in a variant of the TECSP.Assume that instead of weights, with each edge of  is associated a set of colors, also called labels.Let  be the set of all labels of the graph, such that with each edge  ∈  is associated a set of labels   ⊆ .The Generalized Labeled Two Edge Connected Subgraph Problem (GLTECSP) consists in finding a minimum-label spanning 2-edge connected subgraph of , that is a labeled 2-edge connected subgraph with a minimum number of labels.In particular, if each edge  ∈  is associated with exactly one color (label), i.e., |  | = 1, the problem is then called the Labeled Two Edge Connected Subgraph Problem (LTECSP).It is obvious that the GLTECSP is NP-hard.In fact, if with each edge is associated a different single color, that is || = ||, the GLTECSP is equivalent to the TECSP known to be NP-hard [27].
In Figure 1a, we give an example of a graph with six vertices denoted from 1 to 6, and eleven edges denoted   ,  = 1, 2, . . ., 11.There are four labels (colors) in the graph  = {red, blue, green, black}, and with each edge   ,  ∈ {1, 2, . . ., 11} is associated exactly one label from .The objective of the LTECSP is to find a two-edge connected subgraph with a minimum number of labels.Figure 1b illustrates an optimal solution for this example.In fact, in this solution the obtained graph is clearly 2-edge connected, that is between each pair of vertices there are two edge-disjoint paths.Moreover, we are using only two labels, which is obviously the minimum number of labels that we may use in this configuration.
In Figure 2a is illustrated an example for the GLTECSP.It consists of a graph with six nodes, eleven edges and four labels (as previously described).However, in this example, some edges may have more than one label.In Figure 2b an optimal solution of this example is given.The obtained graph is 2-edge connected using two labels, which is the minimum number of labels that can be used.Here, for edge  2 we may retain both labels (blue and green) since this will not change the number of labels in the final solution.
Both the LTECSP and the GLTECSP have several applications in telecommunication networks, electric networks, multimodal transportation networks, where one aims to ensure connectivity by means of homogeneous connections that are generally managed by different and competing companies.This is the case, for example, of telecommunication networks where the different colors can be seen as different operators or different types of media.Recent applications in social networks are depicted in [31], vertices of the graph refers to people, edges represent the relations between them, and different kinds of relationship are labeled with different colors..The paper is organized as follows.In the next section we overview some related works.We mainly distinguish works that are related to network survivability and those that are related to labeled problems in graphs.In Section 3, we propose Integer Programming (IP) formulations for the GLTECSP, a natural formulation based on cuts on the edges, and a compact formulation based on cuts on the colors.In Section 4, we devise Branch-and-Cut algorithms for both formulations and present experimental results held on randomly generated instances.

Survivable network design problem
The TECSP is a classical well known problem in network design.Inspired from the telecommunication context, survivable network design problems have been intensively studied for several years (see [27] for a survey on the problem).Several resolution methods including heuristics, approximation algorithms as well as exact algorithms have been developed to solve various variants of the problem.Consider an undirected graph  = (, ) representing a telecommunication network and a weight function which associates the weight () with every edge  ∈ .With each node  ∈  is associated an integer, denoted by () and called connectivity type of , which can be seen as the minimum number of edges connecting  to the rest of the network.A subgraph  = (,  ) of  where  ⊆  and  ⊆  is said to satisfy the edge-connectivity (resp.node-connectivity) requirement if for every pair of nodes (, ) ∈  ×  , there exist at least (, ) = min{(), ()} edge-disjoint (resp.node-disjoint) paths between  and .Recall that two -paths are edge-disjoint if they have no edge in common, and they are node-disjoint if they have no internal node in common.The edge-connectivity (resp.node-connectivity) condition ensures that the network remains functional (connected) when some equipments fail.In fact, the traffic can still be routed between two nodes  and  when at most (, ) − 1 links, in case of edge-connectivity, and at most (, ) − 1 nodes, in case of node-connectivity, fail.When () = , for every  ∈  , the subgraph  is -edge-connected (resp.-node-connected).In [23], Grötschel et al. address the general survivable network design problem that consists in finding a minimum-weight subgraph of  while satisfying the connectivity requirements.In particular, the -edge-connected subgraph problem, which consists in finding a minimum-weight spanning subgraph of  that is -edge-connected, has been extensively studied, especially in the case of low connectivity requirement ( = 2, i.e., the TECSP).In fact, the 2-connectivity has shown to be a very suitable topology for telecommunication networks.In [28], Mahjoub introduces a general class of valid inequalities for the polytope associated with the TECSP, called  -partition inequalities.Boyd and Hao [4] describe a class of valid inequalities for the TECSP polytope that are "comb"-like inequalities.In [2], Barahona and Mahjoub characterize the TECSP's polytope for the class of Halin graphs.An interesting variant of the problem has also been studied, this is the case when  ∈ {0, 2}.The nodes with () = 2 are called terminals and those with () = 0 are called Steiner nodes.The terminals need high degree of survivability.The Steiner ones are optional nodes, they may be used if they contribute reducing the cost of the network.This problem is known as the Steiner 2-edge connected subgraph problem (STECSP).In [1], Baïou and Mahjoub study the STECSP's polytope.In parallel, Coullard et al. study the node version [11,12].In [11], a linear time algorithm for the problem on some special classes of graphs is proposed.In [12], the authors describe the dominant of the relaxed polytope in a special class of graphs.

Classical labeled problems in graphs
Several classical well-known problems in graphs have been studied within the labeled version.A particular interest has been provided to the Minimum Labeled Spanning Tree Problem (MLSTP).The problem is proved to be NP-hard with different reduction procedures by Chang and Leu [9], and in parallel by Broersma and Li [5].In [7], Cerruli et al. apply several metaheuristic approaches to solve the MLSTP.These metaheuristics are able to improve over existing heuristics presented in the literature.They also provide optimal or close to optimal solutions compared to the exact ones.In [6], Captivo et al. introduce some mixed Integer Programming formulations for the MLSTP and compare the different linear relaxations.In [10], Consoli et al. propose an intelligent-optimization algorithm to solve the MLSTP.This algorithm is obtained by combining the basic Variable Neighbourhood Search heuristic with some features from machine learning, statistics and experimental algorithmics.In [14], the authors study the Generalized MLSTP, that is when multiple labels can be assigned to an edge.The authors propose a new compact integer programming formulation for the GMLSTP and study the associated polytope.Several families of valid inequalities are identified and their facial aspect is investigated.In [16], the authors introduce a new MIP-based metaheuristic for the MLSTP called the multi-start local branching (MSLB).Computational experiments show that the MSLB outperforms the state-of-the-art metaheuristics with respect to optimality and processing times.In [30], Vaisman proposes a new method to solve the MLSTP called the cross-entropy method, which relies on rigorous developments in the fields of information theory and stochastic simulation.Experimentations indicate that the proposed algorithm can obtain optimal or near-optimal solutions while using a reasonable computational time.Several other connectivity-related problems in edge-labeled graph have been also investigated.In [8], Cerulli et al. studied an extension of the MLSTP where the vertices are divided into terminals and Steiner nodes, that is the Steiner Tree Problem with Minimum number of Labels.The problem is solved using several metaheuristics, and numerical results are presented.
Along with the Spanning tree related problems, scientists have also addressed other labeled-graph problems.In [32], the authors study the labeled TSP.They prove that the problem is NP-hard, and present a heuristic and a genetic algorithm to solve the problem.In [26], the authors propose a mathematical model, valid inequalities and polyhedral results for the minimum labeled Hamiltonian cycle problem.The authors also introduce two variants of this problem and devise Branch-and-Cut algorithms for the different variants.In [3], the Minimum Coloring Cut Problem (MCCP) is studied.The MCCP is defined as follows: given a connected graph  with colored edges, find an edge cut  ′ of  (a minimal set of edges whose removal disconnects the graph) such that the number of colors used by the edges in  ′ is minimum.The authors present two approaches based on Variable Neighborhood Search to solve the problem.The presented algorithms are able to find all the optimal solutions described in the literature.In [18], Faria et al. study the maximum labeled cut problem.Given an edge-labeled graph and a parameter , the question is whether or not  has a nontrivial edge cut using at least  labels (colors).The problem of minimum colorful cut and related problems have also been addressed in [15,17,22].
In this work, we address the TECSP in labeled graphs.To the best of our knowledge, the problem has only been initiated in [13,29], and no algorithmic implementations have been considered for it.

IP Formulations
In this section we propose two Integer Programming (IP) formulations for the GLTECSP.The first one is based on edge-cuts and is called edge-cut formulation (ECut) or natural formulation.The second one uses colorcuts and is called color-cut formulation (CCut) or compact formulation.In order to introduce these formulations, we first give some definitions and notations.

Notations
Let  = (, , ) be an edge-labeled undirected graph, where  is a set of labels, such that with each edge  ∈  is associated a subset of labels   ⊆ .For  ⊂  and  ̸ = ∅, let ( ) denote the set of edges of  having one node in  and the other in  =  ∖  .( ) is called a cut or an edge-cut.When  = {},  ∈  , we will write ().

Natural formulation
We define three families of binary decision variables.For each label  ∈ , let   = 1 if label  is selected in the final solution, and 0 if not.For each edge  ∈  and each label  ∈   , let   = 1 if edge  having label  is retained in the final solution, and 0 if not.Finally, for each edge  ∈ , we let   = 1 if edge  is selected in the final solution, and 0 if not. will be called label variables, and  and  design variables.
The GLTECSP is equivalent to the following integer linear program.
(ECut) min ≥ 2 for all  ⊂ , and ) ∈ {0, 1} for all  ∈ , for all  ∈   . (3.10) By the objective function (3.1), the number of labels is minimized.Inequalities (3.2) are called edge-cut inequalities.These ensure that every cut in the graph has a cardinality at least 2, which implies that the solution is 2-edge connected.Note that inequalities (3.2) have an exponential number which means that solving (ECut) to optimality requires their separation which will be detailed in Section 4. Constraints (3.3) are setting the relationship between the design variables   and   .Inequalities (3.4)

Compact formulation
In [13], the author propose a formulation for the -edge connected subgraph problem and the 2-node connected subgraph problem based on the so-called color-cut or colorful cut.In order to adapt this formulation to the GLTECSP, we introduce some definitions and notations that will be useful in the sequel [13].
For a cut ( ) let (( )) = {  :  ∈ ( )} be the set of labels associated with the edges of ( ).Denote by ( ) = (( )) the cut on labels derived from the edge-cut ( ).( ) ⊆  is called color-cut or colorful cut.In the sequel, ({}) will denote the set of edges that have label  ∈ .This will simply be denoted by () for  ∈ .
The GLTECSP is equivalent to the following IP.Obviously, (CCut) has significantly less variables than (ECut), and that is why we will call it compact formulation even though inequalities (3.12) are in exponential number and need to be separated.Note also that (CCut) enables having more than one color per edge in the final solution, which is not the case for the (ECut) formulation due to constraints (3.3).This means that the solution space of (ECut) is included in the (CCut) one.The optimal solution has though always the same number of labels.Moreover, (CCut) would clearly have a better linear relaxation than (ECut), which will be proved by experimentations in the following section.

Experimental results
In this section, we devise Branch-and-Cut algorithms for the edge-cut formulation (ECut) and the color-cut formulation (CCut) previously introduced for the GLTECSP.The Branch-and-Cut algorithms are implemented in C++ using CPLEX 12.101 as a linear solver.Computations are performed on an Intel-Core-i7 2.80 GHz with 16 GB of RAM, running under Linux (Ubuntu 21.0).Tests are computed on complete-graph instances that have been randomly generated, and time limit is set to 1 h.

Branch-and-cut algorithms
The IP formulations proposed for the GLTECSP are given with a huge number of edge-cut inequalities for the (ECut) formulation (resp.color-cut inequalities for the (CCut) formulation).We first consider a restricted version of the linear program corresponding to the (ECut) formuation (resp.the (CCut) formulation).A restricted number of edge-cut inequalities (resp.color-cut inequalities) is generated in the first LP.For the (ECut), we generate only edge-cut inequalities on the vertices.Similarly, only color-cut inequalities on the vertices are generated for the (CCut) formulation.Therefore, the initial linear program for the (ECut) (resp.(CCut)) formulation denoted by LP(ECut) (resp.LP(CCut)) solved in the first step is given by inequalities (3.2) (resp.inequalities (3.12)) written for  = {} for each  ∈  .12 forall the open sub-problem do 13 go to 2; 14 return the best optimal solution of all the sub-problems.
Denote by (, , ) ∈ R  × R  × R × the solution of the linear relaxation LP(ECut).The obtained solution (, , ) is optimal for the restricted LP if and only if it satisfies all the edge-cut inequalities (3.2).In general, this is not the case.Therefore, violated edge-cut inequalities are added to the restricted LP by solving a subproblem called separation problem (see Sect. 4.2 for details).The process is repeated until no more violated inequality is found.The final solution is hence optimal for the linear relaxation LP(ECut).If the solution is integer, then to apply separation only for integer solutions (see Algorithm 2).The separation procedure can be described as follows.As , the obtained optimal solution for the LP(CCut), is integer, we will apply a maximum flow algorithm to look for a violated cut.To this end, we use LEMON2 to compute these minimum cuts.

Computational results
The experimental results are reported in Tables 1 and 2  For each size of | | and ||, we generate 5 random instances.Consequently, each line of the tables reports the average values over the 5 tested instances of the same size.Columns opt-EC (resp.opt-CC) report the number of instances over 5 that have been solved to optimality within the time limit.Overall we tested 61 × 5 instances (61 sizes/lines).
From the results, it is clear that the (CCut) formulation performs better than (ECut) formulation.In fact, over the 61 sizes, the (CCut) formulation was able to solve 58 to optimality.That is to say for 58 lines of the tables, all the 5/5 instances have been solved to optimality.However, the (ECut) formulation left 24 lines incomplete, which means that for each size (line) among the 24, at least one instance could not be solved to optimality within the time limit.And among these 24 lines, 16 have 0/5 solved instances, which says that for all these sizes, the (ECut) formulation was unable to solve the generated instances to optimality.
Along with optimality, the (CCut) formulation outperforms the (ECut) formulation with respect to the computational time.In fact, the results for 39 lines over the 61 show an average computational time less than 2 seconds.In addition, for some sizes where the (ECut) formulation was unable to solve any of the generated instances, the (CCut) formulation provides 5/5 optimal solutions within only some seconds.This is mainly the case of instances with | | = 80 and || ≥ 200.Now concerning the gaps to the root LP relaxation, one can clearly see that the (CCut) gap is slightly better than the (ECut) gap for all the instances (except for line | | = 50 and || = 600).However, for both formulations the gaps are quite high.In fact, these reach an average value of 52% for the (ECut) formulation (see line | | = 80 and || = 100), and 48% for the (CCut) formulation (see line | | = 80 and || = 100).This can be explained by the fact that the average optimal values obtained for all the instances are quite small, not exceeding 14 labels.At this stage, in order to improve the linear relaxations of both (Ecut) and (CCut) formulations, one has to identify new valid inequalities for both, which may tighten the gap and improve the resolution.
We note also that, even having optimality for the majority of the instances, the (CCut) formulation spreads through huge Branch-and-Cut trees compared to the (ECut) formulation.This is clearly depicted in column "Sub-CC" that gives the number of explored subproblems reaching huge values particularly for the instances with | | ∈ {90, 100}.This is mainly due to the fact that the separation phase is ensured only for integer solutions.This is also clear through the number of separated color-cuts that is equal to 0 for 41 out of 61 lines of the tables.As a consequence, improving the separation phase mainly by devising efficient routines has to be considered for future works.

Concluding remarks
In this paper, we study the Generalized Labeled Two Edge Connected Subgraph Problem (GLTECSP) that is a variant of the Two Edge Connected Subgraph Problem, a well-studied problem in network design.In this variant, a set of labels (colors) is associated with each edge in the graph, and the objective is to look for a minimum-label 2-edge connected subgraph.We propose two IP formulations for the GLTECSP, the first is based on edge-cuts and the second uses color-cuts.We discuss the separation routines and devise Branch-and-Cut algorithms for both formulations.The experimentations show that the color-cut formulation performs better than the edge-cut one in terms of linear relaxation and computational time.Several instances that could not be solved within the time limit by the edge-cut formulation were solved to optimality by the color-cut formulation within some seconds.In the future, we aim at improving the linear relaxations for both formulations mainly by adding valid inequalities.In fact, even if the formulations are able to solve the majority of the instances, the value of the gaps to the root bounds is significantly large and needs to be reduced.
are giving the relationship between the design variables   and the label variables   .Constraints (3.5)-(3.7)are the trivial constraints (3.8)-(3.10)are the integrality constraints of the decision variables.
. Entries of the tables are averages values as follows: