Consider the 2-matching problem defined on the complete graph, with edge costs which satisfy the triangle inequality. We prove that the value of a minimum cost 2-matching is bounded above by 4/3 times the value of its...
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Consider the 2-matching problem defined on the complete graph, with edge costs which satisfy the triangle inequality. We prove that the value of a minimum cost 2-matching is bounded above by 4/3 times the value of its linear programming relaxation, the fractional 2-matching problem. This lends credibility to a long-standing conjecture that the optimal value for the traveling salesman problem is bounded above by 4/3 times the value of its linear programming relaxation, the subtour elimination problem.
Let G = (V, E) be a graph. An edge set {uv is-an-element-of E\u is-an-element-of S(i), v is-an-element-of S(j), i not-equal j}, where S1,..., S(k) is a partition of V, is called a multicut with k shores. We investigat...
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Let G = (V, E) be a graph. An edge set {uv is-an-element-of E\u is-an-element-of S(i), v is-an-element-of S(j), i not-equal j}, where S1,..., S(k) is a partition of V, is called a multicut with k shores. We investigate the polytopes MC(k)less-than-or-equal-to (n) and MC(k)greater-than-or-equal-to (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 K(n). 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 MC(k)less-than-or-equal-to (n) and MC(k)greater-than-or-equal-to (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.
In this paper, we give a complete characterization of the class of weighted maximum multiflow problems whose dual polyhedra have bounded fractionality. This is a common generalization of two fundamental results of Kar...
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In this paper, we give a complete characterization of the class of weighted maximum multiflow problems whose dual polyhedra have bounded fractionality. This is a common generalization of two fundamental results of Karzanov. The first one is a characterization of commodity graphs H for which the dual of maximum multiflow problem with respect to H has bounded fractionality, and the second one is a characterization of metrics d on terminals for which the dual of metric-weighed maximum multiflow problem has bounded fractionality. A key ingredient of the present paper is a nonmetric generalization of the tight span, which was originally introduced for metrics by Isbell and Dress. A theory of nonmetric tight spans provides a unified duality framework to the weighted maximum multiflow problems, and gives a unified interpretation of combinatorial dual solutions of several known min-max theorems in the multiflow theory. (C) 2009 Elsevier Inc. All rights reserved.
We study the stable set polytope P(G(n)) for the graph G(n) with n nodes and edges [i, j] with j is an element of {i + 1;i + 2}, i = 1;...;n and where nodes n + 1 and 1 (resp., n + 2 and 2) are identified. This graph ...
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We study the stable set polytope P(G(n)) for the graph G(n) with n nodes and edges [i, j] with j is an element of {i + 1;i + 2}, i = 1;...;n and where nodes n + 1 and 1 (resp., n + 2 and 2) are identified. This graph coincides with the antiweb (W) over bar(n;3). A minimal linear system defining P(G(n)) is determined. The system consists of certain rank inequalities with some number theoretic flavor. A characterization of the vertices of a natural relaxation of P(G(n)) is also given.
Categorical judgement data are analyzed along the lines of random utility theory. A class of orders is introduced (categorical weak orders);their characteristic vectors are regarded as points in a Euclidean space;thei...
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Categorical judgement data are analyzed along the lines of random utility theory. A class of orders is introduced (categorical weak orders);their characteristic vectors are regarded as points in a Euclidean space;their convex hull forms a polytope whose facets are fully characterized. This polytope is shown to correspond to an order polytope. Furthermore, its relation to the biorder polytope is pointed out. The convex representations of a given point of the polytope are discussed. The impact of these results on the methods of analyzing data arising from a categorical judgement procedure is outlined. In particular, some consequences are drawn with respect to the usual evaluation of correlations of such data. (c) 2005 Elsevier Inc. All rights reserved.
Given any family F of valid inequalities for the asymmetric traveling salesman polytope P(G) defined on the complete digraph G, we show that all members of F are facet defining if the primitive members of F (usually a...
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Given any family F of valid inequalities for the asymmetric traveling salesman polytope P(G) defined on the complete digraph G, 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 for P(G) by lifting inequalities that are facet inducing for P(G'), where G' is some induced subgraph of G. 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 of G' with cliques of G 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.
The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the 'Simplicial Steinitz problem'. It is known by an indirect an...
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The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the 'Simplicial Steinitz problem'. It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich-Rubinstein polytopes by showing that the boundary sphere of the KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to the Bier sphere of the associated simplicial complex of "short sets".
The SQAP-polytope was associated to quadratic assignment problems with a certain symmetric objective function structure by Rijal (1995) and Padberg and Rijal (1996). We derive a technique for investigating the SQAP-po...
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The SQAP-polytope was associated to quadratic assignment problems with a certain symmetric objective function structure by Rijal (1995) and Padberg and Rijal (1996). We derive a technique for investigating the SQAP-polytope that is based on projecting the (low-dimensional) polytope into a lower dimensional vector-space, where the vertices have a more convenient coordinate structure. We exploit this technique in order to prove conjectures by Padberg and Rijal on the dimension of the SQAP-polytope as well as on its trivial facets.
Given a rectangle R in the plane and a finite set P of points in its interior, consider the partitions of the surface of R into smeller rectangles. A partition is feasible with respect to P if each point in P lie on t...
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Given a rectangle R in the plane and a finite set P of points in its interior, consider the partitions of the surface of R into smeller rectangles. A partition is feasible with respect to P if each point in P lie on the boundary of some rectangle of the partition. The length of a partition is computed as the sum of the lengths of the line segments defining the boundary of its rectangles. The goal is to find a feasible partition with minimum length. This problem, denoted by RGP, belongs to NP-hard and has application in VLSI design. In this paper we investigate how to obtain exact solutions for the RGP. We introduce two different Integer Programming formulations and carry out a theoretical study to evaluate and compare the strength of their bounds. Computational experiments are reported for Branch-and-Cut and Branch-and-Price algorithms we have implemented for the first and the second formulation, respectively. Randomly generated instances with \P\ less than or equal to 200 are solved exactly. The tests indicate that the size of the instances solved with our algorithms decrease by an order of magnitude in the absence of corectilinear points in P, a special case of RGP whose complexity is still open.
Leontief substitution systems have been studied by economists and operations researchers for many rears. We show how such linear systems are naturally viewed as Leontief substitution flow problems on directed hypergra...
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Leontief substitution systems have been studied by economists and operations researchers for many rears. We show how such linear systems are naturally viewed as Leontief substitution flow problems on directed hypergraphs, and that important solution properties follow from structural characteristics of the hypergraphs. We give a strongly polynomial, non-simplex algorithm for Leontief substitution flow problems that satisfy a gainfree property leading to acyclic extreme solutions. Integrality conditions follow easily from this algorithm. Another structural property, support disjoint reachability, leads to necessary and sufficient conditions for extreme solutions to be binary. In a survey of applications, we show how the Leontief flow paradigm links polyhedral combinatorics, expert systems, mixed integer model formulation, and some problems in graph optimization.
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