Facet-inducing Inequalities Research Articles

Facet separation for disjunctive constraints with network flow representation

We present a novel algorithm for separating facet-inducing inequalities for the convex-hull of the union of polytopes representing a disjunctive constraint of special structure. It is required that the union of polytopes admit a certain network flow representation. The algorithm is based on a new, graph theoretic characterization of the facets of the convex-hull. Moreover, we characterize the family of polytopes that are representable by the networks under consideration. We demonstrate the effectiveness of our approach by computational results on a set of benchmark problems.

The polytope of binary sequences with bounded variation

We investigate the problem of optimizing a linear objective function over the set of all binary vectors of length n with bounded variation, where the latter is defined as the number of pairs of consecutive entries with different value. This problem arises naturally in many applications, e.g., in unit commitment problems or when discretizing binary optimal control problems subject to a bounded total variation. We study two variants of the problem. In the first one, the variation of the binary vector is penalized in the objective function, while in the second one, it is bounded by a hard constraint. We show that the first variant is easy to deal with while the second variant turns out to be more complex, but still tractable. For the latter case, we present a complete polyhedral description of the convex hull of feasible solutions by facet-inducing inequalities and devise an exact linear-time separation algorithm. The proof of completeness also yields a new exact primal algorithm with a running time of O(nlogn), which is significantly faster than the straightforward dynamic programming approach. Finally, we devise a compact extended formulation.A preliminary version of this article has been published in the Proceedings of the 7th International Symposium on Combinatorial Optimization (ISCO 2022) (Buchheim and Hügging, 2022).

Algorithms for the clique problem with multiple-choice constraints under a series–parallel dependency graph

The clique problem with multiple-choice constraints (CPMC), i.e. the problem of finding a k -clique in a k -partite graph with known partition, occurs as a substructure in many real-world applications, in particular scheduling and railway timetabling. Although CPMC is NP-complete in general, it is known to be solvable in polynomial time when the so-called dependency graph of G is a forest. In this article, we focus on the special case CPMCSP, where the dependency graph of G is series–parallel. We give a polynomial-time algorithm for CPMCSP using dynamic programming. Further, we provide some facet-inducing inequalities of the CPMCSP polytope, mainly using properties of the stable set polytope of the complement graph of G . Among these, we give a separation algorithm for the so-called embedded odd-clique-cycle inequalities using dynamic programming. If the number of vertices per subset of the k -partition is bounded, then its runtime is polynomial in the size of the dependency graph.

A polyhedral study of a relaxation of the routing and spectrum allocation problem (Brief Announcement)

The routing and spectrum allocation (RSA) problem arises in the context of flexible grid optical networks, and consists in routing a set of demands through a network while simultaneously assigning a bandwidth to each demand, subject to non-overlapping constraints. One of the most effective integer programming formulations for RSA is the DR-AOV formulation, presented in a previous work. In this work we explore a relaxation of this formulation with a subset of variables from the original formulation, in order to identify valid inequalities that could be useful within a cutting-plane environment for tackling RSA. We present basic properties of this relaxed formulation, we identify several families of facet-inducing inequalities, and we show that they can be separated in polynomial time.

The maximum 2D subarray polytope: Facet-inducing inequalities and polyhedral computations

Given a matrix with real-valued entries, the maximum 2D subarray problem consists in finding a rectangular submatrix with consecutive rows and columns maximizing the sum of its entries. In this work we start a polyhedral study of an integer programming formulation for this problem. We thus define the 2D subarray polytope , explore conditions ensuring the validity of linear inequalities, and provide several families of facet-inducing inequalities. We also report computational experiments assessing the reduction of the dual bound for the linear relaxation achieved by these families of inequalities.

Facet-generating procedures for the maximum-impact coloring polytope

Given two graphs G = ( V , E G ) and H = ( V , E H ) over the same set of vertices and given a set of colors C , the impact on H of a coloring c : V → C of G , denoted I ( c ) , is the number of edges i j ∈ E H such that c ( i ) = c ( j ) . In this setting, the maximum-impact coloring problem asks for a proper coloring c of G maximizing the impact I ( c ) on H . This problem naturally arises in the context of classroom allocation to courses, where it is desirable – but not mandatory – to assign lectures from the same course to the same classroom. In a previous work we identified several families of facet-inducing inequalities for a natural integer programming formulation of this problem. Most of these families were based on similar ideas, leading us to explore whether they can be expressed within a unified framework. In this work we tackle this issue, by presenting two procedures that construct valid inequalities from existing inequalities, based on extending individual colors to sets of colors and on extending edges of G to cliques in G , respectively. If the original inequality defines a facet and additional technical hypotheses are satisfied, then the obtained inequality also defines a facet. We show that these procedures can explain most of the inequalities presented in a previous work, we present a generic separation algorithm based on these procedures, and we report computational experiments showing that this approach is effective.

Facets based on cycles and cliques for the acyclic coloring polytope

A coloring of a graph is an assignment of colors to its vertices such that any two vertices receive distinct colors whenever they are adjacent. An acyclic coloring is a coloring such that no cycle receives exactly two colors, and the acyclic chromatic number χA(G) of a graph G is the minimum number of colors in any such coloring of G. Given a graph G and an integer k, determining whether χA(G) ≤ k or not is NP-complete even for k = 3. The acyclic coloring problem arises in the context of efficient computations of sparse and symmetric Hessian matrices via substitution methods. In a previous work we presented facet-inducing families of valid inequalities based on induced even cycles for the polytope associated to an integer programming formulation of the acyclic coloring problem. In this work we continue with this study by introducing new families of facet-inducing inequalities based on combinations of even cycles and cliques.

Facet-inducing inequalities with acyclic supports for the caterpillar-packing polytope

A caterpillar is a connected graph such that the removal of all its vertices with degree 1 results in a path. Given a graph G, a caterpillar-packing of G is a set of vertex-disjoint (not necessarily induced) subgraphs of G such that each subgraph is a caterpillar. In this work we consider the set of caterpillar-packings of a graph, which corresponds to feasible solutions of the 2-schemes strip cutting problem with a sequencing constraint (2-SSCPsc) presented by Rinaldi and Franz (Eur. J. Oper. Res. 183 (2007) 1371–1384). Facet-preserving procedures have been shown to be quite effective at explaining the facet-inducing inequalities of the associated polytope, so in this work we continue this issue by exploring such procedures for valid inequalities with acyclic supports. In particular, the obtained results are applicable when the underlying graph is a tree.

The 2D Subarray Polytope

Given a d-dimensional array, the maximum subarray problem consists in finding an axis-parallel slice of the array maximizing the sum of its entries. In this work we start a polyhedral study of a natural integer programming formulation for this problem when d = 2. Such an exploration is motivated by the need of solving large-scale instances of the rectilinear picture compression problem (RPC), a problem which arises in different scenarios. The obtained results can be useful to solve the column generation phase of a branch and price approach for RPC, a technique that applies naturally to this problem. We thus define the 2D subarray polytope, explore conditions ensuring the validity of linear inequalities, and provide several families of facet-inducing inequalities.

The minimum chromatic violation problem: A polyhedral approach

In this paper we define a generalization of the classical vertex coloring problem of a graph, where some pairs of adjacent vertices can be assigned to the same color. We call weak an edge connecting two such vertices. We look for a coloring of the graph minimizing the number of weak edges having its endpoints assigned to the same color. This problem is called the minimum chromatic violation problem (MCVP). We present an integer programming formulation for this problem and provide an initial polyhedral study of the polytope arising from this formulation. We give partial characterizations of facet-inducing inequalities and we show how facets from different instances of MCVP are related. We then introduce general lifting procedures which generate (sometimes facet-inducing) valid inequalities from generic valid inequalities. We exhibit several facet-inducing families arising from these procedures and we present a family of facet-inducing inequalities which is not obtained from the prior lifting procedures, associated with certain substructures in the given graph. Finally, we analyze the extreme case of all weak edges and its relationship with the well-known k-partition problem.

Exact algorithms for the minimum cost vertex blocker clique problem

We study the minimum cost vertex blocker clique problem (CVCP), which is defined as follows. Given a simple undirected graph with weights and costs on its vertices and r > 0, detect a minimum cost subset of vertices to be removed such that the weight of any clique in the remaining graph is at most r. We propose a new characterization of the set of feasible solutions that results in new linear 0–1 programming formulations solved by lazy-fashioned branch-and-cut algorithms. A new set of facet-inducing inequalities for the convex hull of feasible solutions to CVCP are also identified. We also propose new combinatorial bounding schemes and employ them to develop the first combinatorial branch-and-bound algorithm for this problem. The computational performance of these exact algorithms is studied on a test-bed of randomly generated graphs, and real-life instances.

Balanced vehicle routing: Polyhedral analysis and branch-and-cut algorithm

This paper studies a variant of the unit-demand Capacitated Vehicle Routing Problem, namely the Balanced Vehicle Routing Problem, where each route is required to visit a maximum and a minimum number of customers. A polyhedral analysis for the problem is presented, including the dimension of the associated polyhedron, description of several families of facet-inducing inequalities and the relationship between these inequalities. The inequalities are used in a branch-and-cut algorithm, which is shown to computationally outperform the best approach known in the literature for the solution of this problem.

On the combinatorics of the 2-class classification problem

A set of points X=XB∪XR⊆Rd is linearly separable if the convex hulls of XB and XR are disjoint, hence there exists a hyperplane separating XB from XR. Such a hyperplane provides a method for classifying new points, according to which side of the hyperplane the new points lie. When such a linear separation is not possible, it may still be possible to partition XB and XR into prespecified numbers of groups, in such a way that every group from XB is linearly separable from every group from XR. We may also discard some points as outliers, and seek to minimize the number of outliers necessary to find such a partition. Based on these ideas, Bertsimas and Shioda proposed the classification and regression by integer optimization (CRIO) method in 2007. In this work we explore the integer programming aspects of the classification part of CRIO, in particular theoretical properties of the associated formulation. We are able to find facet-inducing inequalities coming from the stable set polytope, hence showing that this classification problem has exploitable combinatorial properties.

The sociotechnical teams formation problem: a mathematical optimization approach

Based on the sociometric analysis of social networks, we introduce the sociotechnical teams formation problem (STFP). Given a group of individuals with different skill-sets and a social network that captures the mutual affinity between them, the problem consists in finding a set of pairwise disjoint teams, as harmonious as possible, with a minimum specified number of individuals per team per skill. We prove that STFP is $$\mathcal {NP}$$-hard and propose an integer linear programming formulation. We show several classes of facet-inducing inequalities for the corresponding polytope. Computational experiments performed on a set of 120 test instances attest the efficiency of a solution method based on the formulation strengthened by valid inequalities and on a simulated annealing algorithm used to provide good initial feasible solutions.

Facet-inducing inequalities and a cut-and-branch for the bandwidth coloring polytope based on the orientation model

The bandwidth coloring problem (BCP) is a generalization of the well-known vertex coloring problem (VCP), asking colors to be assigned to vertices of a graph such that the absolute difference between the colors assigned to adjacent vertices is greater than or equal to a weight associated to the edge connecting them. In this work we present an integer programming formulation for BCP based on the orientation model for VCP. We present two families of valid inequalities for this formulation, show that they induce facets of the associated polytope, and report computational experience suggesting that these families are useful in practice.

The minimum chromatic violation problem: a polyhedral study

We propose a generalization of the k-coloring problem, namely the minimum chromatic violation problem (MCVP). Given a graph G=(V,E), a set of weak edgesF⊂E and a set of colors C, the MCVP asks for a |C|-coloring of the graph G′=(V,E\\F) minimizing the number of weak edges with both endpoints receiving the same color. We present an integer programming formulation for this problem and provide an initial polyhedral study of the polytopes arising from this formulation. We give partial characterizations of facet-inducing inequalities and we show how facets from weaker and stronger instances of MCVP (i.e., more/less weak edges) are related. We then introduce a general lifting procedure which generates (sometimes facet-inducing) valid inequalities from generic valid inequalities and we present several facet-inducing families arising from this procedure. Finally, we present another family of facet-inducing inequalities which is not obtained from the prior lifting procedure.

On facet-inducing inequalities for combinatorial polytopes

One of the central questions of polyhedral combinatorics is the question of the algorithmic relationship between the vertex and facet descriptions of convex polytopes. From the standpoint of combinatorial optimization, the main reason for the actuality of this question is the possibility of applying the methods of convex analysis to solving the extremal combinatorial problems. In this paper, we consider the combinatorial polytopes of a sufficiently general form. We obtain a few of necessary conditions and a sufficient condition for a supporting inequality of a polytope to be a facet inequality and give an illustration of the use of the developed technology to the polytope of some graph approximation problem.

The caterpillar-packing polytope

A caterpillar is a connected graph such that the removal of all its vertices with degree 1 results in a path. Given a graph G, a caterpillar-packing ofG is a set of vertex-disjoint (not necessarily induced) subgraphs of G such that each subgraph is a caterpillar. In this work we consider the set of caterpillar-packings of a graph, which corresponds to feasible solutions of the 2-schemes strip cutting problem with a sequencing constraint (2-SSCPsc) presented by F. Rinaldi and A. Franz in 2007. We study the polytope associated with a natural integer programming formulation of this problem. We explore basic properties of this polytope, including a lifting lemma and several facet-preserving operations on the graph. These results allow us to introduce several families of facet-inducing inequalities.

The generalized independent set problem: Polyhedral analysis and solution approaches

In the Generalized Independent Set Problem, we are given a graph, a revenue for each vertex, and a set of removable edges with associated removal costs, and we seek to find an independent set that maximizes the net benefit, i.e., the difference between the revenues collected for the vertices in the independent set and the costs incurred for any removal of edges with both endpoints in the independent set. We study the polyhedron associated with a 0–1 linear programming formulation of the Generalized Independent Set Problem, deriving a number of facet-inducing inequalities, and we develop linear programming based heuristics to obtain high-quality solutions in a short amount of time. We also develop a heuristic method based on an unconstrained 0–1 quadratic programming formulation of the Generalized Independent Set Problem. In an extensive computational study, we assess the performance of these heuristics in terms of quality and efficiency. The best heuristic is then used to produce an initial solution for a branch-and-cut algorithm which uses some of the proposed facet-inducing inequalities.

Exploring the disjunctive rank of some facet-inducing inequalities of the acyclic coloring polytope

In a previous work we presented six facet-inducing families of valid inequalities for the polytope associated to an integer programming formulation of the acyclic coloring problem. In this work we study their disjunctive rank, as defined by [E. Balas, S. Ceria and G. Cornuejols, Math. Program. 58 (1993) 295–324]. We also propose to study a dual concept, which we call the disjunctive anti-rank of a valid inequality.