In this paper we investigate the convex hull of single node variable upper-bound flow models with allowed configurations. Such a model is defined by a set X-rho(Z) = {(x, z) is an element of R-n x Z vertical bar Sigma...
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In this paper we investigate the convex hull of single node variable upper-bound flow models with allowed configurations. Such a model is defined by a set X-rho(Z) = {(x, z) is an element of R-n x Z vertical bar Sigma(n)(j=1) x(j)rho d, 0 <= x(j) <= u(j)z(j), j = 1,..., n}, where rho is one of of <=, = >=, and Z subset of {0. 1}(n) consists of the allowed configurations. We consider the case when Z consists of affinely independent vectors. Under this assumption, a characterization of the non-trivial facets of the convex hull of X-rho(Z) for each relation p is provided, alone with polynomial time separation algorithms. Applications in scheduling and network design are also discussed. (c) 2006 Elsevier B.V. All rights reserved.
This paper addresses the problem of improving the polyhedral representation of a certain class of machine scheduling problems. Despite the poor polyhedral representation of many such problems in general, it is shown t...
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This paper addresses the problem of improving the polyhedral representation of a certain class of machine scheduling problems. Despite the poor polyhedral representation of many such problems in general, it is shown that notably tighter linear programming representations can be obtained for many important models. In particular, we study the polyhedral structure of two different mixed-integer programming formulations of the flow shop scheduling problem with sequence-dependent setup times, denoted by SDST flow shop. The first is related to the asymmetric traveling salesman problem (ATSP) polytope. The second is less common and is derived from a model proposed by Srikar and Ghosh based on the linear ordering problem (LOP) polytope. The main contribution of this work is the proof that any facet-defining inequality (facet) of either of these polytopes (ATSP and LOP) induces a facet for the corresponding SDST flow shop polyhedron. The immediate benefit of this result is that all developments to date on facets and valid inequalities for both the ATSP and the LOP can be applied directly to the machine scheduling polytope. In addition, valid mixed-integer inequalities based on variable upper-bound flow inequalities for either model are developed as well. The derived cuts are evaluated within a branch-and-cut framework.
A simple 2-matching in a graph is a subgraph whose connected components are nontrivial paths and cycles. A simple 2-matching is called 1-restricted if each connected component has two or more edges. In this paper we c...
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A simple 2-matching in a graph is a subgraph whose connected components are nontrivial paths and cycles. A simple 2-matching is called 1-restricted if each connected component has two or more edges. In this paper we consider the problem of finding maximum weight 1-restricted simple 2-matchings (which is a relaxation of the traveling salesman problem). We present an integer programming formulation for this problem, characterize the extreme points of the linear programming relaxation, and characterize the graphs for which the linear programming relaxation has all integral extreme points. We show how to recognize these graphs in polynomial time. We also introduce a new class of blossom-type inequalities that tighten the general linear programming relaxation. A complete description of the convex hull of 1-restricted simple 2-matchings is unknown.
We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices xx(H), where the elements of x is an element of C-n aremth unit roots. These polytopes have appli-cations in MAX-3-CUT, digital commun...
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We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices xx(H), where the elements of x is an element of C-n aremth unit roots. These polytopes have appli-cations in MAX-3-CUT, digital communication technology, angular synchronizationand more generally, complex quadratic programming. Form=2, the complex cutpolytope corresponds to the well-known cut polytope. We generalize valid cuts for this polytope to cuts for any complex cut poly tope with finitem>2 and provide a frame work to compare them. Further, we consider a second semi definite lifting of thecomplex cut polytope for m=infinity. This lifting is proven to be equivalent to other complex Lasserre-type liftings of the same order proposed in the literature, while beingof smaller size. Our theoretical findings are supported by numerical experiments onvarious optimization problems.
A path-block cycle is a graph that consists of several cycles that all intersect in a common subset of nodes. The associated path-block-cycle inequalities are valid, and sometimes facet-defining, inequalities for poly...
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A path-block cycle is a graph that consists of several cycles that all intersect in a common subset of nodes. The associated path-block-cycle inequalities are valid, and sometimes facet-defining, inequalities for polytopes in connection with graph partitioning problems and corresponding multicut problems. Special cases of the inequalities were introduced by De Souza and Laurent (1995) and shown to be facet-defining for the equicut polytope. Generalizations of these inequalities were shown by Ferreira et al. (1996) to be valid for node-capacitated graph partitioning polytopes on general graphs. This paper considers the special case of the inequalities, where all cycles intersect in two nodes, and establishes conditions under which these inequalities induce facets of node-capacitated multicut polytopes and bisection cut polytopes. These polytopes are associated with simple versions of the node-capacitated graph partitioning and bisection problems, where all node weights are assumed to be 1. (C) 2017 Elsevier B.V. All rights reserved.
In this paper we prove two lifting theorems for the clique partitioning polytope, which provide sufficient conditions for a valid inequality to be facet-defining. In particular, if a valid inequality defines a facet o...
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In this paper we prove two lifting theorems for the clique partitioning polytope, which provide sufficient conditions for a valid inequality to be facet-defining. In particular, if a valid inequality defines a facet of the polytope corresponding to the complete graph K-m on m vertices, it defines a facet for the polytope corresponding to K-n for all n>m. This answers a question raised by Grotschel and Wakabayashi. Further, for the case of arbitrary graphs, we characterize when the so-called 2-partition inequalities define facets. (C) 1999 Elsevier Science B.V. All rights reserved.
The natural linear programming formulation of the maximum s-t-flow problem in path variables has a dual linear program whose underlying polyhedron is the dominant P(s-t-cut)(up arrow) of the s-t-cut polytope. We prese...
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The natural linear programming formulation of the maximum s-t-flow problem in path variables has a dual linear program whose underlying polyhedron is the dominant P(s-t-cut)(up arrow) of the s-t-cut polytope. We present a complete characterization of P(s-t-cut)(up arrow) with respect to vertices, facets, and adjacency.
Acoloringof a graph is an assignment of colors to its vertices such that any two vertices receive distinct colors whenever they are adjacent. Anacyclic coloringis a coloring such that no cycle receives exactly two col...
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Acoloringof a graph is an assignment of colors to its vertices such that any two vertices receive distinct colors whenever they are adjacent. Anacyclic coloringis a coloring such that no cycle receives exactly two colors, and theacyclic chromatic number chi(A)(G) of a graphGis the minimum number of colors in any such coloring ofG. Given a graphGand an integerk, determining whether chi(A)(G) <= kor not is NP-complete even fork = 3. The acyclic coloring problem arises in the context of efficient computations of sparse and symmetric Hessian matricesviasubstitution 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.
The matroid parity (MP) problem is a powerful (and NP-hard) extension of the matching problem. Whereas matching polytopes are well understood, little is known about MP polytopes. We prove that, when the matroid is lam...
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The matroid parity (MP) problem is a powerful (and NP-hard) extension of the matching problem. Whereas matching polytopes are well understood, little is known about MP polytopes. We prove that, when the matroid is laminar, the MP polytope is affinely congruent to a perfect b-matching polytope. From this we deduce that, even when the matroid is not laminar, every Chvatal-Gomory cut for the MP polytope can be derived as a {0, 1/2}-cut from a laminar family of rank constraints. We also prove a negative result concerned with the integrality gap of two linear relaxations of the MP problem. (C) 2020 Elsevier B.V. All rights reserved.
This paper investigates a technique of building up discrete relaxations of combinatorial optimization problems. To establish such a relaxation we introduce a transformation technique aggregation - that allows one to r...
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This paper investigates a technique of building up discrete relaxations of combinatorial optimization problems. To establish such a relaxation we introduce a transformation technique aggregation - that allows one to relax an integer program by means of another integer program. We show that knapsack and set packing relaxations give rise to combinatorial cutting planes in a simple and straightforward way. The constructions are algorithmic. (C) 2001 Elsevier Science B.V. All rights reserved.
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