Given a directed graph with capacities and interdiction costs associated with its arcs, the maximum flow blocker problem (MFBP) asks to find a minimum-cost subset of arcs to be removed from the graph in such a way that the remaining maximum-flow value does not exceed a given threshold. The MFBP has applications in telecommunication networks and in the monitoring of civil infrastructures, among others. We propose an integer linear programming formulation (ILP) with an exponential number of constraints, called Benders cut, for the MFBP. Accordingly, we derive a branch-and-cut algorithm to optimally solve the problem. Preliminary experimental results are reported to assess performance of the formulation and more precisely to determine the dimension of the problem that could be solved to proven optimality.

In this paper, we study the Constrained-Routing and Spectrum Assignment (C-RSA) problem. Consider an undirected, loopless, and connected graph${G}$. Let$\mathbb{S}$be an optical spectrum of available contiguous frequency slots, and${K}$be a set of traffic demands. The C-RSA is to assign for each demand${k\,\in \,K}$a path in${G}$between its origin-destination nodes, and an interval of contiguous frequency slots in$\mathbb{S}$while respecting some technological constraints, and optimizing some linear objective function. First, we propose an extended integer linear programming formulation for the C-RSA. A column generation algorithm is developed to solve its linear relaxation. We also describe several valid inequalities for the polytope associated with this formulation, and address the related separation problems. Using these results, we derive a Branch-and-Cut-and-Price algorithm, along with computational results are presented using large-scale instances.

The Variable-Sized Bin Packing Problem with Color Constraints (VSBPP-CC) is a generalization of the classical one-dimensional Bin Packing Problem, where bins of different capacities are available for packing a set of items each characterized by a weight and a color. The objective is to pack all the items while minimizing the total residual capacity, and such that each bin contains at most two different colors. In this paper we consider a special case of VSBPP-CC where each color is assigned to only one item. We first describe the problem, its practical context and survey related works. We then propose several original mathematical formulations for the problem. Preliminary computational results show the efficiency of our formulations, mainly the so-called matching formulation, in solving CSPLIB-based instances.

In this paper we study a network design problem that consists in finding a minimum weight subgraph containing solutions for multiple Steiner Traveling Salesman Problems with Order constraints. We propose a layered compact ILP formulation for the problem. Experimental results show that it is reasonably effective and can solve to optimality medium-sized instances. Lage-scale instances are more difficult, and does not reach optimal solutions within a time limit of 3 hours. In order to improve our formulation, we investigate valid inequalities efficiency using a column-generation-based approach.

We show that Karger's randomized contraction method (SODA 93) can be adapted to multiobjective global minimum cut problems with a constant number of edge or node budget constraints to give efficient algorithms. For global minimum cuts with a single edge-budget constraint, our extension of the randomized contraction method has running time tilde{O}(n^3) in an n-node graph improving upon the best-known randomized algorithm with running time tilde{O}(n^4) due to Armon and Zwick (Algorithmica 2006). Our analysis also gives a new upper bound of O(n^3) for the number of optimal solutions for a single edge-budget min cut problem. For the case of (k-1) edge-budget constraints, the extension of our algorithm saves a logarithmic factor from the best-known randomized running time of O(n^{2k} log^3 n). A main feature of our algorithms is to adaptively choose, at each step, the appropriate cost function used in the random selection of edges to be contracted. For the global min cut problem with a constant number of node budgets, we give a randomized algorithm with running time tilde{O}(n^2), improving the current best determinisitic running time of O(n^3) due to Goemans and Soto (SIAM Journal on Discrete Mathematics 2013). Our method also shows that the total number of distinct optimal solutions is bounded by O(n^2) as in the case of global min-cuts. Our algorithm extends to the node-budget constrained global min cut problem excluding a given sink with the same running time and bound on number of optimal solutions, again improving upon the best-known running time by a factor of O(n). For node-budget constrained problems, our improvements arise from incorporating the idea of merging any infeasible super-nodes that arise during the random contraction process. In contrast to cuts excluding a sink, we note that the node-cardinality constrained min-cut problem containing a given source is strongly NP-hard using a reduction from graph bisection.

In this paper we discuss a variant of the well-known k-separator problem. Given a simple graph G = (V ∪ T, E) with V ∪ T the set of vertices, where T is a set of distinguished vertices called terminals, and E a set of edges, the multi-terminal vertex separator problem consists in partitioning V ∪T into k+1 subsets {S, V1, ..., Vk} such that the size of S is minimum, each subset Vicontains exactly one terminal and no vertex in Viis adjacent to a vertex in Vj. Three extended formulations are proposed for the problem. We develop Branch-and-Price algorithms for the two first formulations and a Branch-and-Cut-and-Price algorithm for the third one. Some experimental results are also discussed.

The paper deals with the Knapsack Problem with conflicts, also known as the Disjunctively Constrained Knapsack Problem (DCKP). The conflicts are represented by a graph whose vertices are the items such that adjacent items cannot be packed in the knapsack simultaneously. We consider a classical formulation for the DCKP, study the polytope associated with this formulation and investigate the facial aspect of its basic constraints. We then present some valid inequalities. Based on this study, we discuss separation routines of the valid inequalities and devise a Branch-and-Cut algorithm. Preliminary results on a set of problem instances are also given.

Network design problems have been largely studied in the last decades due to the ubiquity of IT communication in our daily life. We address in this paper the k-edge-connected hop-constrained network design problem (kHNDP) which is known to be NP-hard. In this paper, we present a hybrid parallel approach for solving the kHNDP based on a Lagrangian relaxation algorithm, a greedy algorithm, and a genetic algorithm. Computational results obtained with our algorithms are compared with those from CPLEX.

In this article, we study the k-edge-connected L-hop-constrained network design problem. Given a weighted graph , a set D of pairs of nodes, two integers and , the problem consists in finding a minimum weight subgraph of G containing at least k edge-disjoint paths of length at most L between every pair . We consider the problem in the case where L = 2, 3 and . We first discuss integer programming formulations introduced in the literature. Then, we introduce new integer programming formulations for the problem that are based on the transformation of the initial undirected graph into directed layered graphs. We present a theoretical comparison of these formulations in terms of LP-bound. Finally, these formulations are tested using CPLEX and compared in a computational study for k = 3, 4, 5. © 2015 Wiley Periodicals, Inc. NETWORKS, 67(2), 148–169 2016

Given a weighted undirected graph G with a set of pairs of terminals (si, ti), , and an integer , the two node-disjoint hop-constrained survivable network design problem is to find a minimum weight subgraph of G such that between every si and ti there exist at least two node-disjoint paths of length at most L. This problem has applications in the design of survivable telecommunication networks with QoS-constraints. We discuss this problem from a polyhedral point of view. We present several classes of valid inequalities along with necessary and/or sufficient conditions for these inequalities to be facet defining. We also discuss separation routines for these classes of inequalities, and propose a Branch-and-Cut algorithm for the problem when L = 3, as well as some computational results. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 67(4), 316–337 2016

A star forest is a collection of vertex-disjoint trees of depth at most 1, and its size is the number of leaves in all its components. A spanning star forest of a given graph G is a spanning subgraph of G that is also star forest. The spanning star forest problem (SSFP for short) [10] is to find maximum size spanning star forest of given graph. Let define some graph G = (V;E), to every star forest we associate a vector xF. xF(e) = 1 if e ∈ F and xF(e) = 0 otherwise. xFis the incident vector of spanning star forest F. The convex hull of all spanning star forest incident vectors is called a spanning star forest polytope, denoted SFP(G). In this paper we are mainly interested on complete characterization of SFP(G).

Given an indirected graph G = (V;E), a Vertex k-Separator is a subset of the vertex set V such that, when the separator is removed from the graph, the remaining vertices can be partitioned into k subsets that are pairwise edge-disconnected. In this paper we focus on the Balanced Vertex k-Separator Problem, i.e., the problem of finding a minimum cardinality separator such that the sizes of the resulting disconnected subsets are balanced. We present a compact Integer Linear Programming formulation for the problem, and present a polyhedral study of the associated polytope. We also present an Exponential-Size formulation, for which we derive a column generation and a branching scheme. Preliminary computational results are reported comparing the performance of the two formulations on a set of benchmark instances.

Given G = (V;E) an undirected graph and a nonnegative cost function c : E → ℚ, the 2-edge connected spanning subgraph problem (TECSP for short) is to find a two-edge connected subgraph HP = (V; F) of G with minimum cost (i.e., c(F) = Σe∈Fc(e) is minimum). If c(e) > 0 for all e ∈ E then every optimal solution for TECSP is an inclusionwise minimal two-edge connected subgraph. In this paper we provide preliminary results, from a polyhedral point of view, concerning the inclusionwise minimal solutions of TECSP. This problem is clearly NP-Hard. We propose an ILP formulation for the problem and study the associated polytope for the wheels. Morever, we describe some valid inequalities and propose a branch-and-cut algorithm for the problem.

The motivation behind this study is the essential need for survivability in the telecommunications networks. An optical signal should find its destination even if the network experiences an occasional fiber cut. We consider the design of a two-level survivable telecommunications network. Terminals compiling the access layer communicate through hubs forming the backbone layer. To hedge against single link failures in the network, we require the backbone subgraph to be two-edge connected and the terminal nodes to connect to the backbone layer in a dual-homed fashion, i.e., at two distinct hubs. The underlying design problem partitions a given set of nodes into hubs and terminals, chooses a set of connections between the hubs such that the resulting backbone network is two-edge connected, and for each terminal chooses two hubs to provide the dual-homing backbone access. All of these decisions are jointly made based on some cost considerations. We give alternative formulations using cut inequalities, compare these formulations, provide a polyhedral analysis of the small-sized formulation, describe valid inequalities, study the associated separation problems, and design variable fixing rules. All of these findings are then utilized in devising an efficient branch-and-cut algorithm to solve this network design problem.

Abstract The hop‐constrained survivable network design problem consists of finding a minimum cost subgraph containing K edge‐disjoint paths with length at most H joining each pair of vertices in a given demand set. When all demands have a common vertex, the instance is said to be rooted. We propose a new extended formulation for the rooted case, called hop‐level multicommodity flow (MCF), that can be significantly stronger than the previously known formulations, at the expense of having a larger number of variables and constraints, growing linearly with the number of edges and demands and quadratically with H . However, for the particular case where H = 2, it can be specialized into a very compact and efficient formulation. Even when H = 3, hop‐level‐MCF can still be quite efficient and it has solved several instances from the literature for the first time. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

Abstract The survivable hierarchical telecommunications network design problem consists of locating concentrators, assigning user nodes to concentrators, and linking concentrators in a reliable backbone network. In this article, we study this problem when the backbone is 2‐edge connected and when user nodes are linked to concentrators by a point‐to‐point access network. We formulate this problem as an integer linear program and present a facial study of the associated polytope. We describe valid inequalities and give sufficient conditions for these inequalities to be facet defining. We investigate the computational complexity of the corresponding separation problems. We propose some reduction operations to speed up the separation procedures. Finally, we devise a branch‐and‐cut algorithm based on these results and present the outcome of a computational study. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

Abstract In this article, we consider the k‐edge connected subgraph problem from a polyhedral point of view. We introduce further classes of valid inequalities for the associated polytope and describe sufficient conditions for these inequalities to be facet defining. We also devise separation routines for these inequalities and discuss some reduction operations that can be used in a preprocessing phase for the separation. Using these results, we develop a Branch‐and‐Cut algorithm and present some computational results. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

In this paper, we present procedures to obtain facet-defining inequalities for the independence system polytope. These procedures are defined for inequalities which are not necessarily rank inequalities. We illustrate the use of these procedures by deriving strong valid inequalities for the acyclic induced subgraph, triangle free induced subgraph, bipartite induced subgraph, and knapsack polytopes. Finally, we derive a new family of facet-defining inequalities for the independence system polytope by adding a set of edges to antiwebs.

This paper deals with the survivable network design problem where each node v has a connectivity type $r(v)$ equal to 1 or 2, and the survivability conditions require the existence of at least $\min\{r(s),r(t)\}$ edge-disjoint paths for all distinct nodes s and t. We consider the polytope given by the trivial and cut inequalities together with the partition inequalities. More precisely, we study some structural properties of this polytope which leads us to give some sufficient conditions for this polytope to be integer in the class of series-parallel graphs. With both separation problems for the cut and partition inequalities being polynomially solvable, we then obtain a polynomial time algorithm for the (1,2)-survivable network design problem in a subclass of series-parallel graphs including the outerplanar graph class. We also introduce a new class of facet-defining inequalities for the polytope associated to the (1,2)-survivable network design problem.

Abstract This article deals with the Two‐edge connected Hop‐constrained Network Design Problem (or THNDP for short). Given a weighted graphG= (N,E), an integerL≥ 2, and a subset of pairs of nodesD, the problem consists of finding the minimum cost subgraph inGcontaining at least two edge‐disjoint paths of at mostLhops between all the pairs inD. First, we show that the THNDP is stronglyNP‐hard even when the demands inDare rooted at some nodesand the costs are unitary. However, if the graph is complete, we prove that the problem in this case can be solved in polynomial time. We give an integer programming formulation of the problem in the space of the design variables whenL= 2, 3. Then we study the associated polytope. In particular, we consider the case where all the pairs of nodes ofDare rooted at a nodes. We give several classes of valid inequalities along with necessary and/or sufficient conditions for these inequalities to be facet defining. We also derive separation routines for these inequalities. We finally develop a branch‐and‐cut algorithm based on these results and discuss some computational results forL= 2, 3. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 49(1), 116–133 2007

Abstract In this article, we consider the two 4‐hop‐constrained paths problem, which consists, given a graph G = ( N , E ) and two nodes s , t ∈ N , of finding a minimum cost subgraph in G containing at least two node‐ (resp., edge‐) disjoint paths of length at most 4 between s and t . We give integer programming formulations, in the space of the design variables, for both the node and edge versions of this problem. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 49(2), 135–144 2007

Given a graph $G=(V,E)$ with nonnegative weights on the edges, the maximum induced bipartite subgraph problem (MIBSP) is to find a maximum weight bipartite subgraph $(W,E[W])$ of G. Here $E[W]$ is the edge set induced by W. An edge subset $F\subseteq E$ is called independent if there is an induced bipartite subgraph of G whose edge set contains F. Otherwise, it is called dependent. In this paper we characterize the minimal dependent sets, that is, the dependent sets that are not contained in any other dependent set. Using this, we give an integer linear programming formulation for MIBSP in the natural variable space, based on an associated class of valid inequalities called dependent set inequalities. Moreover, we show that the minimum dependent set problem with nonnegative weights can be reduced to the minimum circuit problem in a directed graph, and can then be solved in polynomial time. This yields a polynomial-time separation algorithm for the dependent set inequalities as well as a polynomial-time cutting plane algorithm for solving the linear relaxation of the problem. We also discuss some polyhedral consequences.

Abstract For the past few decades, combinatorial optimization techniques have been shown to be powerful tools for formulating and solving optimization problems arising from practical situations. In particular, many network design problems have been formulated as combinatorial optimization problems. With the advances of optical technologies and the explosive growth of the Internet, telecommunication networks have seen an important evolution and therefore designing survivable networks has become a major objective for telecommunication operators. Over the past years, much research has been carried out to devise efficient methods for survivable network models, and particularly cutting plane based algorithms. In this paper, we attempt to survey some of these models and the optimization methods used for solving them. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 1–21 2005

Given a graph G with distinguished nodes s and t, a cost on each edge of G, and a fixed integer L \geq 2, the two edge-disjoint hop-constrained paths problem is to find a minimum cost subgraph such that between s and t there exist at least two edge-disjoint paths of length at most L. In this paper, we consider that problem from a polyhedral point of view. We give an integer programming formulation for the problem when L = 2,3. An extension of this result to the more general case where the number of required paths is arbitrary and L = 2,3 is also given. We discuss the associated polytope, P(G,L), for L = 2,3. In particular, we show in this case that the linear relaxation of P(G,L), Q(G,L), given by the trivial, the st-cut, and the so-called L-path-cut inequalities, is integral. As a consequence, we obtain a polynomial time cutting plane algorithm for the problem when L = 2,3. We also give necessary and sufficient conditions for these inequalities to define facets of P(G,L) for L \geq 2 when G is complete. We finally investigate the dominant of P(G,L) and give a complete description of this polyhedron for L \geq 2 when P(G,L) = Q(G,L).

Given a graph G=(V,E) with weights on its edges and a set of specified nodes $S\subseteq V$, the Steiner 2-edge survivable network problem is to find a minimum weight subgraph of G such that between every two nodes of S there are at least two edge-disjoint paths. This problem has applications to the design of reliable communication and transportation networks. In this paper, we give a complete linear description of the polytope associated with the solutions to this problem when the underlying graph is series-parallel. We also discuss related polyhedra.

Given a directed graph D that has a two-vertex cut, this paper describes a technique to derive a linear system that defines the acyclic subgraph polytope of D from systems related to the pieces. It also gives a technique to describe facets of this polytope by composition of facets for the pieces. The authors prove that, if the systems for the pieces are totally dual integral (TDI), then the system for D is also. The authors prove that the “cycle inequalities” form a TDI system for any orientation of $K_5$. These results are combined with Lucchesi–Younger theorem and a theorem of Wagner to prove that, for graphs with no $K_{3,3} $ minor, the cycle inequalities characterize the acyclic subgraph polytope and form a TDI system. This shows that, for this class of graphs, the cardinality of a minimum feedback set is equal to the maximum number of arc disjoint cycles. For planar graphs, this is a consequence of the Lucchesi–Younger theorem.

Let $P( G )$ be the balanced induced subgraph polytope of G. If G has a two-node cutset, then G decomposes into $G_1 $ and $G_2$. It is shown that $P( G )$ can be obtained as a projection of a polytope defined by a system of inequalities that decomposes into two pieces associated with $G_1 $ and $G_2$. The problem max $cx,x \in P( G )$ is decomposed in the same way. This is applied to series-parallel graphs to show that, in this case, $P( G )$ is a projection of a polytope defined by a system with $O( n )$ inequalities and $O( n )$ variables, where n is the number of nodes in G. Also for this class of graphs, an algorithm is given that finds a maximum weighted balanced induced subgraph in $O( n\log n )$ time. This approach is also used to obtain composition of facets of $P( G )$. Analogous results are presented for acyclic induced subgraphs.

A graph G with a two-node cutset decomposes into two pieces. A technique to describe the stable set polytope for G based on stable set polytopes associated with the pieces is studied. This gives a way to characterize this polytope for classes of graphs that can be recursively decomposed. This also gives a procedure to describe new facets of this polytope. A compact system for the stable set problem in series-parallel graphs is derived. This technique is also applied to characterize facet-defining inequalities for graphs with no $K_5 \backslash e$ minor. The stable set problem is polynomially solvable for this class of graphs. Compositions of h-perfect graphs are also studied.

The authors characterize the stable set polytope for graphs that do not have a 4-wheel as a minor. The authors prove that the nontrivial facets are either “edge” inequalities or can be obtained by composing “odd cycles” and “subdivisions of $K_4 $.” By adding some extra variables, it is shown that the stable set problem for these graphs can be formulated as a linear program of polynomial size.