Facet-inducing Inequalities Research Articles

The caterpillar-packing polytope

A caterpillar is a 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 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.

A branch-and-cut algorithm for the profitable windy rural postman problem

In this paper we study the profitable windy rural postman problem. This is an arc routing problem with profits defined on a windy graph in which there is a profit associated with some of the edges of the graph, consisting of finding a route maximizing the difference between the total profit collected and the total cost. This problem generalizes the rural postman problem and other well-known arc routing problems and has real-life applications, mainly in snow removal operations. We propose here a formulation for the problem and study its associated polyhedron. Several families of facet-inducing inequalities are described and used in the design of a branch-and-cut procedure. The algorithm has been tested on a large set of benchmark instances and compared with other existing algorithms. The results obtained show that the branch-and-cut algorithm is able to solve large-sized instances optimally in reasonable computing times.

A branch-and-cut algorithm for the Orienteering Arc Routing Problem

In arc routing problems, customers are located on arcs, and routes of minimum cost have to be identified. In the Orienteering Arc Routing Problem (OARP), in addition to a set of regular customers that have to be serviced, a set of potential customers is available. From this latter set, customers have to be chosen on the basis of an associated profit. The objective is to find a route servicing the customers which maximize the total profit collected while satisfying a given time limit on the route. In this paper, we describe large families of facet-inducing inequalities for the OARP and present a branch-and-cut algorithm for its solution. The exact algorithm embeds a procedure which builds a heuristic solution to the OARP on the basis of the information provided by the solution of the linear relaxation. Extensive computational experiments over different sets of OARP instances show that the exact algorithm is capable of solving to optimality large instances, with up to 2000 vertices and 14,000 arcs, within 1h and often within a few minutes.

A polyhedral study of the maximum stable set problem with weights on vertex-subsets

Given a graph G=(V,E), a family of nonempty vertex-subsets S⊆2V, and a weight w:S→R+, the maximum stable set problem with weights on vertex-subsets consists in finding a stable set I of G maximizing the sum of the weights of the sets in S that intersect I. This problem arises within a natural column generation approach for the vertex coloring problem. In this work we perform an initial polyhedral study of this problem, by introducing a natural integer programming formulation and studying the associated polytope. We address general facts on this polytope including some lifting results, we provide connections with the stable set polytope, and we present three families of facet-inducing inequalities.

Minimum edge blocker dominating set problem

This paper introduces and studies the minimum edge blocker dominating set problem (EBDP), which is formulated as follows. Given a vertex-weighted undirected graph and r > 0, remove a minimum number of edges so that the weight of any dominating set in the remaining graph is at least r. Dominating sets are used in a wide variety of graph-based applications such as the analysis of wireless and social networks. We show that the decision version of EBDP is NP-hard for any fixed r > 0. We present an analytical lower bound for the value of an optimal solution to EBDP and formulate this problem as a linear 0–1 program with a large number of constraints. We also study the convex hull of feasible solutions to EBDP and identify facet-inducing inequalities for this polytope. Furthermore, we develop the first exact algorithm for solving EBDP, which solves the proposed formulation by a branch-and-cut approach where nontrivial constraints are applied in a lazy fashion. Finally, we also provide the computational results obtained by using our approach on a test-bed of randomly generated instances and real-life power-law graphs.

The Team Orienteering Arc Routing Problem

The team orienteering arc routing problem (TOARP) is the extension to the arc routing setting of the team orienteering problem. In the TOARP, in addition to a possible set of regular customers that have to be serviced, another set of potential customers is available. Each customer is associated with an arc of a directed graph. Each potential customer has a profit that is collected when it is serviced, that is, when the associated arc is traversed. A fleet of vehicles with a given maximum traveling time is available. The profit from a customer can be collected by one vehicle at most. The objective is to identify the customers that maximize the total profit collected while satisfying the given time limit for each vehicle. In this paper we propose a formulation for this problem and study a relaxation of its associated polyhedron. We present some families of valid and facet-inducing inequalities that we use in the implementation of a branch-and-cut algorithm for the resolution of the problem. Computational experiments are run on a large set of benchmark instances.

Unbounded convex sets for non-convex mixed-integer quadratic programming

This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some other well-studied convex sets. Several classes of valid and facet-inducing inequalities are also derived.

A polyhedral approach to the single row facility layout problem

The single row facility layout problem (SRFLP) is the NP-hard problem of arranging facilities on a line, while minimizing a weighted sum of the distances between facility pairs. In this paper, a detailed polyhedral study of the SRFLP is performed, and several huge classes of valid and facet-inducing inequalities are derived. Some separation heuristics are presented, along with a primal heuristic based on multi-dimensional scaling. Finally, a branch-and-cut algorithm is described and some encouraging computational results are given.

Multi-depot Multiple TSP: a polyhedral study and computational results

We study the Multi-Depot Multiple Traveling Salesman Problem (MDMTSP), which is a variant of the very well-known Traveling Salesman Problem (TSP). In the MDMTSP an unlimited number of salesmen have to visit a set of customers using routes that can be based on a subset of available depots. The MDMTSP is an NP-hard problem because it includes the TSP as a particular case when the distances satisfy the triangular inequality. The problem has some real applications and is closely related to other important multi-depot routing problems, like the Multi-Depot Vehicle Routing Problem and the Location Routing Problem. We present an integer linear formulation for the MDMTSP and strengthen it with the introduction of several families of valid inequalities. Certain facet-inducing inequalities for the TSP polyhedron can be used to derive facet-inducing inequalities for the MDMTSP. Furthermore, several inequalities that are specific to the MDMTSP are also studied and proved to be facet-inducing. The partial knowledge of the polyhedron has been used to implement a Branch-and-Cut algorithm in which the new inequalities have been shown to be very effective. Computational results show that instances involving up to 255 customers and 25 possible depots can be solved optimally using the proposed methodology.

Combinatorial properties and further facets of maximum edge subgraph polytopes

Abstract Given a graph G and an integer k, the maximum edge subgraph problem consists in finding a k-vertex subset of G such that the number of edges within the subset is maximum. This NP-hard problem arises in the analysis of cohesive subgroups in social networks. In this work we study the polytope P ( G , k ) associated with a straightforward integer programming formulation of the maximum edge subgraph problem. We characterize the graph generated by P ( G , k ) and give a tight bound on its diameter. We give a complete description of P ( K 1 n , k ) , where K 1 n is the star on n + 1 vertices, and we conjecture a complete description of P ( m K 2 , k ) , where m K 2 is the graph composed by m disjoint edges. Finally, we introduce three families of facet-inducing inequalities for P ( G , k ) , which generalize known families of valid inequalities for this polytope.

Disjunctive ranks and anti-ranks of some facet-inducing inequalities of the acyclic coloring polytope

Abstract A coloring of a graph G is an assignment of colors to the vertices of G such that any two vertices receive distinct colors whenever they are adjacent. An acyclic coloring of G is a coloring such that no cycle of G receives exactly two colors, and the acyclic chromatic number χ A ( 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 χ A ( 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 this work we study the disjunctive rank of six facet-inducing families of valid inequalities for the polytope associated to a natural integer programming formulation of the acyclic coloring problem. We also introduce the concept of disjunctive anti-rank and study the anti-rank of these families.

A polyhedral study of the single-item lot-sizing problem with continuous start-up costs

Abstract In this work we consider the single-item single-machine lot-sizing problem with continuous start-up costs. A continuous start-up cost is generated in a period whenever there is a nonzero production in the period and the production capacity in the previous period is not saturated. This concept of start-up does not correspond to the standard (discrete) start-up considered in previous models, thus motivating a polyhedral study of this problem. We introduce a natural integer programming formulation for this problem, we study some general properties and facet-inducing inequalities of the associated polytope, and we state relationships with known lotsizing polytopes.

New results on the Windy Postman Problem

In this paper, we study the Windy Postman Problem (WPP). This is a well-known Arc Routing Problem that contains the Mixed Chinese Postman Problem (MCPP) as a special case. We extend to arbitrary dimension some new inequalities that complete the description of the polyhedron associated with the Windy Postman Problem over graphs with up to four vertices and ten edges. We introduce two new families of facet-inducing inequalities and prove that these inequalities, along with the already known odd zigzag inequalities, are Chvátal–Gomory inequalities of rank at most 2. Moreover, a branch-and-cut algorithm that incorporates two new separation algorithms for all the previously mentioned inequalities and a new heuristic procedure to obtain upper bounds are presented. Finally, the performance of a branch-and-cut algorithm over several sets of large WPP and MCPP instances, with up to 3,000 nodes and 9,000 edges (and arcs in the MCPP case), shows that, to our knowledge, this is the best algorithm to date for the exact resolution of the WPP and the MCPP.

Projecting an Extended Formulation for Mixed-Integer Covers on Bipartite Graphs

We consider the mixed-integer version of bipartite vertex cover. This is equivalent to a mixed-integer network dual model, introduced recently, that generalizes several mixed-integer sets arising in production planning. We derive properties of inequalities that are valid for the convex hull of the mixed-integer bipartite covers by projecting an extended formulation onto the space of the original variables. This permits us to give a complete description of the facet-inducing inequalities of the double mixing set and of the continuous mixing set with flows, two mixed-integer sets that generalize several models studied in the literature.

A polyhedral study of the maximum edge subgraph problem

Abstract The study of cohesive subgroups is an important aspect of social network analysis. Cohesive subgroups are studied using different relaxations of the notion of clique in a graph. For instance, given a graph and an integer k , the maximum edge subgraph problem consists in finding a k -vertex subset such that the number of edges within the subset is maximum. This work proposes a polyhedral approach for this NP-hard problem. We study the polytope associated to an integer programming formulation of the problem, present several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families.

Facet-inducing inequalities for chromatic scheduling polytopes based on covering cliques

Chromatic scheduling polytopes arise as solution sets of the bandwidth allocation problem in certain radio access networks supplying wireless access to voice/data communication networks to customers with individual communication demands. This bandwidth allocation problem is a special chromatic scheduling problem; both problems are N P -complete and, furthermore, there exist no polynomial-time algorithms with a fixed quality guarantee for them. As algorithms based on cutting planes are shown to be successful for many other combinatorial optimization problems, the goal is to apply such methods to the bandwidth allocation problem. For that, knowledge on the associated polytopes is required. The present paper contributes to this issue, introducing new classes of valid inequalities based on variations and extensions of the covering-clique inequalities presented in [J. Marenco, Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems, Ph.D. Thesis, Universidad de Buenos Aires, Argentina, 2005; J. Marenco, A. Wagler, Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems, Annals of Operations Research 150-1 (2007) 159–175]. We discuss conditions ensuring that these inequalities define facets of chromatic scheduling polytopes, and we show that the associated separation problems are N P -complete.

The Football Pool Polytope

Abstract The football pool problem asks for the minimun number of bets on the result on n football matches ensuring that some bet correctly predicts the outcome of at least n − 1 of them. This combinatorial problem has proven to be extremely difficult, and is open for n ⩾ 6 . Integer programming techniques have been applied to this problem in the past but, in order to tackle the open cases, a deep knowledge of the polytopes associated with the integer programs modeling this problem is required. In this work we address this issue, by defining and studying the football pool polytope in connection with a natural integer programming formulation of the football pool problem. We explore the basic properties of this polytope and present several classes of facet-inducing valid inequalities over natural combinatorial structures in the original problem.

The Windy General Routing Polyhedron: A Global View of Many Known Arc Routing Polyhedra

The windy postman problem consists of finding a minimum cost traversal of all of the edges of an undirected graph with two costs associated with each edge, representing the costs of traversing it in each direction. In this paper we deal with the windy general routing problem (WGRP), in which only a subset of edges must be traversed and a subset of vertices must be visited. This is also an ${\it NP}$-hard problem that generalizes many important arc routing problems (ARPs) and has some interesting real-life applications. Here we study the description of the WGRP polyhedron, for which some general properties and some large families of facet-inducing inequalities are presented. Moreover, since the WGRP contains many well-known routing problems as special cases, this paper also provides a global view of their associated polyhedra. Finally, for the first time, some polyhedral results for several ARPs defined on mixed graphs formulated by using two variables per edge are presented.

Polynomial-Time Separation of a Superclass of Simple Comb Inequalities

The comb inequalities are a well-known class of facet-inducing inequalities for the traveling salesman problem, defined in terms of certain vertex sets called the handle and the teeth. We say that a comb inequality is simple if the following holds for each tooth: Either the intersection of the tooth with the handle has cardinality one, or the part of the tooth outside the handle has cardinality one, or both. The simple comb inequalities generalize the classical 2-matching inequalities of Edmonds [Edmonds, J. 1965. Maximum matching and a polyhedron with 0–1 vertices. J. Res. Nat. Bur. Standards 69B 125–130] and also the so-called Chvátal comb inequalities. In 1982, Padberg and Rao [Padberg, M. W., M. R. Rao. 1982. Odd minimum cut-sets and b-matchings. Math. Oper. Res. 7 67–80] gave a polynomial-time separation algorithm for the 2-matching inequalities, i.e., an algorithm for testing if a given fractional solution to an LP relaxation violates a 2-matching inequality. We extend this significantly by giving a polynomial-time separation algorithm for a class of valid inequalities which includes all simple comb inequalities.

Zigzag inequalities: a new class of facet-inducing inequalities for Arc Routing Problems

In this paper we introduce a new class of facet-inducing inequalities for the Windy Rural Postman Problem and the Windy General Routing Problem. These inequalities are called Zigzag inequalities because they cut off fractional solutions containing a zigzag associated with variables with 0.5 value. Two different types of inequalities, the Odd Zigzag and the Even Zigzag inequalities, are presented. Finally, their application to other known Arc Routing Problems is discussed.