New Classes Of Facets Research Articles

New classes of Facets for complementarity knapsack problems

The complementarity knapsack problem (CKP) is a knapsack problem with real-valued variables and complementarity conditions between pairs of variables. We extend the polyhedral studies of de Farias et al. for CKP, by proposing three new families of cutting-planes that are obtained from a combinatorial concept known as pack. Sufficient conditions for these inequalities to be facet-defining, based on the new concept of a maximal switching pack, are also provided. Moreover, we answer positively a conjecture by de Farias et al. about the separation complexity of the inequalities introduced in their work, and propose efficient separation algorithms for our newly defined cutting-planes.

The opportunistic replacement problem: theoretical analyses and numerical tests

We consider a model for determining optimal opportunistic maintenance schedules with respect to a maximum replacement interval. This problem generalizes that of Dickman et al. (J Oper Res Soc India 28:165–175, 1991) and is a natural starting point for modelling replacement schedules of more complex systems. We show that this basic opportunistic replacement problem is NP-hard, that the convex hull of the set of feasible replacement schedules is full-dimensional, that all the inequalities of the model are facet-inducing, and present a new class of facets obtained through a {0,1/2}-Chvatal–Gomory rounding. For costs monotone with time, a class of elimination constraints is introduced to reduce the computation time; it allows maintenance only when the replacement of at least one component is necessary. For costs decreasing with time, these constraints eliminate non-optimal solutions. When maintenance occasions are fixed, the remaining problem is stated as a linear program and solved by a greedy procedure. Results from a case study on aircraft engine maintenance illustrate the advantage of the optimization model over simpler policies. We include the new class of facets in a branch-and-cut framework and note a decrease in the number of branch-and-bound nodes and simplex iterations for most instance classes with time dependent costs. For instance classes with time independent costs and few components the elimination constraints are used favorably. For fixed maintenance occasions the greedy procedure reduces the computation time as compared with linear programming techniques for all instances tested.

Polyhedral structure of the 4‐node network design problem

AbstractThis article studies the polyhedral structure of the 4‐node network design problem (NDP). Using a theorem from the previous work of this author, the facets of the 4‐node NDP can be translated into facets of larger size problems. The knowledge of complete polyhedral description of the 4‐node NDP is important because it implies complete knowledge of 4‐partition‐based facets of larger NDPs. After reviewing the previously known facets of the 4‐node NDP, a new class of facets is derived. An enumerative methodology is presented for determining whether a given set of inequalities provides a complete polyhedral description. By implementing this methodology in a computer code, it is determined that the known facets of the 4‐node NDP indeed provide a complete polyhedral description of the problem. Working of the proof methodology is illustrated with examples, and the results of the computer enumeration are reported. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009

New facets for the planar subgraph polytope

AbstractThis study describes certain facet classes for the planar subgraph polytope. These facets are extensions of Kuratowski facets and are of the form 2x(U) + x(E(G)\U) ≤ 2|U| + |E(G)\U| − 2 where the edge set U varies and can be empty. Two of the new types of facets complete the class of extended subdivision facets, explored by Jünger and Mutzel. In addition, the other types of facets consist of a new class of facets for the polytope called 3‐star subdivisions. It is also shown that the extended and 3‐star subdivision facets are also equivalent to members of the class of facets with coefficients in {0, 1, 2} for the set covering polytope. Computational results displaying the effectiveness of the facets in a branch‐and‐cut scheme for the maximum planar subgraph problem are presented. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

On the asymmetric representatives formulation for the vertex coloring problem

We consider the vertex coloring problem, which can be stated as the problem of minimizing the number of labels that can be assigned to the vertices of a graph G such that each vertex receives at least one label and the endpoints of every edge are assigned different labels. In this work, the 0–1 integer programming formulation based on representative vertices is revisited to remove symmetry. The previous polyhedral study related to the original formulation is adapted and generalized. New versions of facets derived from substructures of G are presented, including cliques, odd holes and anti-holes and wheels. In addition, a new class of facets is derived from independent sets of G. Finally, a comparison with the independent sets formulation is provided.

New classes of facets of the cut polytope and tightness of [formula omitted] Bell inequalities

The Grishukhin inequality is a facet of the cut polytope CUT 7 □ of the complete graph K 7 , for which no natural generalization to a family of inequalities has previously been found. On the other hand, the I mm 22 Bell inequalities of quantum information theory, found by Collins and Gisin, can be seen as valid inequalities of the cut polytope CUT □ ( ∇ K m , m ) of the complete tripartite graph ∇ K m , m = K 1 , m , m . They conjectured that they are facet inducing. We prove their conjecture by relating the I mm 22 inequalities to a new class of facets of CUT N □ that are a natural generalization of the Grishukhin inequality. An important component of the proof is the use of a method called triangular elimination, introduced by Avis, Imai, Ito and Sasaki, for producing facets of CUT □ ( ∇ K m , m ) from facets of CUT N □ .

IFORS' Operational Research Hall of Fame

International Transactions in Operational ResearchVolume 13, Issue 2 p. 169-174 Free Access IFORS' Operational Research Hall of Fame Egon Balas, Egon BalasSearch for more papers by this author Egon Balas, Egon BalasSearch for more papers by this author First published: 09 March 2006 https://doi.org/10.1111/j.1475-3995.2006.00541.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Selected Original Works Balas, E., 1965. An additive algorithm for solving linear programs with zero–one variables. Operations Research 13, 517– 546. Balas, E., 1969. Machine sequencing via disjunctive graphs: an implicit enumeration algorithm. Operations Research 17, 941– 957. Balas, E., 1970. Machine sequencing: disjunctive graphs and degree-constrained subgraphs. Naval Research Logistics Quarterly 17, 1– 9. Balas, E., 1974. Disjunctive programming: properties of the convex hull of feasible points. MSRR No. 348, Carnegie Mellon University, July. Also 1998. Discrete Applied Mathematics 89, 1– 44. Balas, E., 1989. The prize collecting traveling salesman problem. Networks 19, 621– 636. Balas, E., 1995. The prize collecting traveling salesman problem: II. Networks 25, 199– 216. Balas, E., Adams, J., Zawack, D., 1988. The shifting bottleneck procedure for job shop scheduling. Management Science 34, 391– 401. Balas, E., Ceria, S., Cornuéjols, G., 1993. A lift-and-project cutting plane algorithm for mixed 0–1 programs. Mathematical Programming 58, 295– 324. Balas, E., Fischetti, M., 1993. A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets. Mathematical Programming 58, 325– 352. Balas, E., Fischetti, M., 1999. Lifted cycle inequalities for the asymmetric traveling salesman problem. Mathematics of Operations Research 24, 273– 292. Balas, E., Fischetti, M., Pulleyblank, W.R., 1995. The precedence-constrained asymmetric traveling salesman polytope. Mathematical Programming 68, 241– 265. Balas, E., Pulleyblank, W.R., 1983. The perfectly matchable subgraph polytope of a bipartite graph. Networks 13, 495– 516. Balas, E., Vazacopoulos, A., 1998. Guided local search with shifting bottleneck for job shop scheduling. Management Science 44, 262– 275. Biographical Works Balas, E., 2000. Will to Freedom: A Perilous Journey through Fascism and Communism. Syracuse University Press, Syracuse, NY. Balas, E., 2002. Some thoughts on the development of integer programming during my research career. Lecture delivered upon receiving the EURO Gold Medal, July 9, 2001, Rotterdam. European Journal of Operational Research 141, 1– 7. Volume13, Issue2March 2006Pages 169-174 ReferencesRelatedInformation

A new class of facets for the Latin square polytope

Latin squares of order n have a 1–1 correspondence with the feasible solutions of the 3-index planar assignment problem ( 3 PAP n ). In this paper, we present a new class of facets for the associated polytope, induced by odd-hole inequalities.

On the asymmetric representatives formulation for the vertex coloring problem

We consider the vertex coloring problem, which can be stated as the problem of minimizing the number of labels that can be assigned to the vertices of a graph G such that each vertex receives at least one label and the endpoints of every edge are assigned different labels. In this work, the 0-1 integer programming formulation based on representative vertices is revisited to remove symmetry. The previous polyhedral study related to the original formulation is adapted and generalized. New versions of facets derived from substructures of G are presented, including cliques, odd holes and anti-holes and wheels. In addition, a new class of facets is derived from independent sets of G. Finally, a comparison with the independent sets formulation is provided.

Adapting polyhedral properties from facility to hub location problems

We examine the feasibility polyhedron of the uncapacitated hub location problem (UHL) with multiple allocation, which has applications in the fields of air passenger and cargo transportation, telecommunication and postal delivery services. In particular we determine the dimension and derive some classes of facets for this polyhedron. We develop a general rule about lifting facets from the uncapacitated facility location problem to UHL. Using this lifting procedure we derive a new class of facets for UHL which dominates the inequalities in the original formulation. Thus we obtain a new formulation of the UHL whose constraints are all facet-defining.

Separating Maximally Violated Comb Inequalities in Planar Graphs

The Traveling Salesman Problem (TSP) is a benchmark problem in combinatorial optimization. It was one of the very first problems used for developing and testing approaches to solving large integer programs, including cutting plane algorithms and branch-and-cut algorithms. Much of the research in this area has been focused on finding new classes of facets for the TSP polytope, and much less attention has been paid to algorithms for separating from these classes of facets. In this paper, we consider the problem of finding violated comb inequalities. If there are no violated subtour constraints in a fractional solution of the TSP, a comb inequality may not be violated by more than 1. Given a fractional solution in the subtour elimination polytope whose graph is planar, we either find a violated comb inequality or determine that there are no comb inequalities violated by 1. Our algorithm runs in O(n + MC(n)) time, where MC(n) is the time to compute a cactus representation of all minimum cuts of a weighted planar graph on n vertices.

The general routing problem polyhedron: Facets from the RPP and GTSP polyhedra

In this paper we study the polyhedron associated with the General Routing Problem (GRP). This problem, first introduced by Orloff in 1974, is a generalization of both the Rural Postman Problem (RPP) and the Graphical Traveling Salesman Problem (GTSP) and, thus, is NP -hard. We describe a formulation of the problem such that from every non-trivial facet-inducing inequality for the RPP and GTSP polyhedra, we obtain facet-inducing inequalities for the GRP polyhedron. We describe a new family of facet-inducing inequalities for the GRP, the honeycomb constraints, which seem to be very useful for solving GRP and RPP instances. Finally, new classes of facets obtained by composition of facet-inducing inequalities are presented.

Some new classes of facets for the equicut polytope

Given a graph G = ( V, E), a cut in G that partitions V into two sets with ⌊ 1 2 ¦V¦⌋ and ⌈ 1 2 ¦V¦⌉ nodes is called an equicut. Suppose that there are weights assigned to the edges in E. The problem of finding a minimum weight equicut in G is known to be NP-hard. The equicut polytope is defined as the convex hull of the incidence vectors of the equicuts in G. In this paper we describe several new classes of facets for the equicut polytope; they arise as various generalizations of an inequality based on a cycle introduced by Conforti et al. (1990). Most of our inequalities have the interesting feature that their support graphs are planar but for some of them both planarity and connectivity properties are lost. Finally we show how our results can be applied to obtain new classes of facets for the cut polytope.

A simultaneous lifting strategy for identifying new classes of facets for the Boolean quadric polytope

We develop a framework for characterizing classes of facets for the Boolean quadric polytope obtainable through a simultaneous lifting procedure. In particular, we begin with a class of product-form facets that subsume Padberg's clique, cut, and generalized cut inequality facets. By applying the proposed general approach to this class of facets, we derive a specially structured polyhedron whose vertices describe all facets that are simultaneous liftings of these facets. We identify specific classes of vertices for this polyhedron to reveal a new class of facets for the quadric polytope. Such an approach can be applied to lifting other facets, as well as to analyze other combinatorial optimization problems.

The inequicut cone

Given a complete graph K n on n nodes and a subset S of nodes, the cut δ( S) defined by S is the set of edges of K n with exactly one endnode in S. A cut δ( S) is an equicut if |S|=⌊ n 2 ⌋ or ⌈ n 2 ⌉ and an inequicut, otherwise. The cut cone C n (or inequicut cone IC n ) is the cone generated by the incidence vectors of all cuts (or inequicuts) of K n . The equicut polytope EP n , studied by Conforti et al. (1990), is the convex hull of the incidence vectors of all equicuts. We prove that IC n and EP n ‘inherit’ all facets of the cut cone C n , namely, that every facet of the cut cone C n yields (by zero-lifting) a facet of the inequicut cone IC m for n<⌊ m 2 ⌋ and of EP m for m odd, m⩾2 n+1. We construct several new classes of facets, not arising from C n , for the inequicut cone IC n and we describe its facial structure for n⩽7.

A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets

Given any family ℱ of valid inequalities for the asymmetric traveling salesman polytopeP(G) defined on the complete digraphG, we show that all members of ℱ are facet defining if the primitive members of ℱ (usually a small subclass) are. Based on this result we then introduce a general procedure for identifying new classes of facet inducing inequalities forP(G) by lifting inequalities that are facet inducing forP(G′), whereG′ is some induced subgraph ofG. Unlike traditional lifting, where the lifted coefficients are calculated one by one and their value depends on the lifting sequence, our lifting procedure replaces nodes ofG′ with cliques ofG and uses closed form expressions for calculating the coefficients of the new arcs, which are sequence-independent. We also introduce a new class of facet inducing inequalities, the class of SD (source-destination) inequalities, which subsumes as special cases most known families of facet defining inequalities.

Clique-Web Facets for Multicut Polytopes

Let G = (V, E) be a graph. An edge set {uv ∈ E|u ∈ Si, v ∈ Sj, i ≠ j}, where S1, …, Sk is a partition of V, is called a multicut with k shores. We investigate the polytopes MCk≤(n) and MCk≥(n) that are defined as the convex hulls of the incidence vectors of all multicuts with at most k shores and at least k shores, respectively, of the complete graph Kn. We introduce a large class of inequalities, called clique-web inequalities, valid for these polytopes, and provide a quite complete characterization of those clique-web inequalities that define facets of MCk≤(n) and MCk≥(n). Using general facet manipulation techniques like collapsing and node splitting we construct further new classes of facets for these multicut polytopes. We also exhibit a class of clique-web facets for which the separation problem can be solved in polynomial time.

Max-cut in circulant graphs

We study the max-cut problem in circulant graphs Cn,r, where Cn,r is a graph whose edge set consists of a cycle of length n and all the vertex pairs of distance r on the cycle. An efficient solution of the problem is obtained so that we show that there is always a maximum cut of a particular shape, called a t-regular cut. The number of edges of a t-regular cut can easily be computed. This gives an O(r log2n) time algorithm for the max-cut.We present also some new classes of facets of the bipartite subgraph polytope and the cut polytope, which are spanned by t-regular cuts.

Facets for the cut cone II: Clique-web inequalities

We study new classes of facets for the cut coneC n generated by the cuts of the complete graph onn vertices. This cone can also be interpreted as the cone of all semi-metrics onn points that are isometricallyl 1-embeddable and, in fact, the study of the facets of the cut polytope is in some sense equivalent to the study of the facets ofC n . These new facets belong to the class of clique-web inequalities which generalize the hypermetric and cycle inequalities as well as the bicycle odd wheel inequalities.

The max-cut problem and quadratic 0–1 optimization; polyhedral aspects, relaxations and bounds

Given a graphG, themaximum cut problem consists of finding the subsetS of vertices such that the number of edges having exactly one endpoint inS is as large as possible. In the weighted version of this problem there are given real weights on the edges ofG, and the objective is to maximize the sum of the weights of the edges having exactly one endpoint in the subsetS. In this paper, we consider the maximum cut problem and some related problems, likemaximum-2-satisfiability, weighted signed graph balancing. We describe the relation of these problems to the unconstrained quadratic 0–1 programming problem, and we survey the known methods for lower and upper bounds to this optimization problem. We also give the relation between the related polyhedra, and we describe some of the known and some new classes of facets for them.