Facets of the linear ordering polytope: A unification for the fence family through weighted graphs (original) (raw)
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arXiv (Cornell University), 2020
Ordering polytopes have been instrumental to the study of combinatorial optimization problems arising in a variety of fields including comparative probability, computational social choice, and group decision-making. The weak order polytope is defined as the convex hull of the characteristic vectors of all binary orders on alternatives that are reflexive, transitive, and total. By and large, facet defining inequalities (FDIs) of this polytope have been obtained through simple enumeration and through connections with other combinatorial polytopes. This paper derives five new large classes of FDIs by utilizing the equivalent representation of a weak order as a ranking of alternatives that allows ties; this connection simplifies the construction of valid inequalities, and it enables groupings of characteristic vectors into useful structures. We demonstrate that a number of FDIs previously obtained through enumeration are actually special cases of the large classes. This work also introduces novel construction procedures for generating affinely independent members of the identified ranking structures. Additionally, it states two conjectures on how to derive many more large classes of FDIs using the featured techniques.
Exploring the disjunctive rank of some facet-inducing inequalities of the acyclic coloring polytope
RAIRO - Operations Research, 2016
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. Cornuéjols, 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.
Weighted graphs defining facets: A connection between stable set and linear ordering polytopes
Discrete Optimization, 2009
A graph is α-critical if its stability number increases whenever an edge is removed from its edge set. The class of α-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of is the finite basis theorem for α-critical graphs of a fixed defect. The class of α-critical graphs is also of interest for at least two topics of polyhedral studies. First, Chvátal (1975) shows that each α-critical graph induces a rank inequality which is facet-defining for its stable set polytope. Investigating a weighted generalization, Lovász (2000, 2001) introduce critical facet-graphs (which again produce facet-defining inequalities for their stable set polytopes) and they establish a finite basis theorem. Second, Koppen (1995) describes a construction that delivers from any αcritical graph a facet-defining inequality for the linear ordering polytope. handle the weighted case and thus define facet-defining graphs. Here we investigate relationships between the two weighted generalizations of α-critical graphs. We show that facet-defining graphs (for the linear ordering polytope) are obtainable from 1critical facet-graphs (linked with stable set polytopes). We then use this connection to derive various results on facet-defining graphs, the most prominent one being derived from Lipták and Lovász's finite basis theorem for critical facet-graphs. At the end of the paper we offer an alternative proof of Lovász's finite basis theorem for α-critical graphs.
Proving Facetness of Valid Inequalities for the Clique Partitioning Polytope
In this paper we prove two lifting theorems for the clique partitioning problem. Each of these theorems implies that if a valid inequality satis es certain conditions, then it de nes a facet of the clique partitioning polytope. In particular if a valid inequality de nes a facet of the polytope corresponding to the graph K m , i.e. the complete graph on m vertices, it de nes a facet for the polytope corresponding to K n for all n > m. This answers a question raised by Gr otschel and Wakabayashi.
An Approval-Voting Polytope for Linear Orders
Journal of Mathematical Psychology, 1997
A probabilistic model of approval voting on n alternatives generates a collection of probability distributions on the family of all subsets of the set of alternatives. Focusing on the size-independent model proposed by Falmagne and Regenwetter, we recast the problem of characterizing these distributions as the search for a minimal system of linear equations and inequalities for a specific convex polytope. This approval-voting polytope, with n ! vertices in a space of dimension 2 n , is proved to be of dimension 2 n &n&1. Several families of facet-defining linear inequalities are exhibited, each of which has a probabilistic interpretation. Some proofs rely on special sequences of rankings of the alternatives. Although the equations and facet-defining inequalities found so far yield a complete minimal description when n 4 (as indicated by the PORTA software), the problem remains open for larger values of n. ] 1997 Academic Press article no. MP971155
Gap Inequalities for the Cut Polytope
European Journal of Combinatorics, 1996
We introduce a new class of inequalities valid for the cut polytope, which we call gap inequalities. Each gap inequality is given by a nite sequence of integers, whose \gap" is de ned as the smallest discrepancy arising when decomposing the sequence into two parts as equal as possible. Gap inequalities include the hypermetric inequalities and the negative type inequalities, which have been extensively studied in the literature. They are also related to a positive semide nite relaxation of the max-cut problem. A natural question is to decide for which integer sequences the corresponding gap inequalities de ne facets of the cut polytope. For this property, we present a set of necessary and su cient conditions in terms of the root patterns and of the rank of an associated matrix. We also prove that there is no facet de ning inequality with gap greater than one and which is induced by a sequence of integers using only two distinct values.
On the p‐median polytope and the odd directed cycle inequalities: Oriented graphs
Networks, 2018
We study the classical linear programing relaxation of the ‐median problem, together with the so‐called “odd directed cycle inequalities.” We characterize in terms of forbidden subgraphs, the oriented graphs for which this system of inequalities defines an integral polytope. This completes the study started in Baïou and Barahona (2016), where oriented graphs with no triangles were treated.
On the 0, 1 facets of the set covering polytope
Mathematical Programming, 1989
In this paper,j we consider: inequalities of t-G form ",.-equals 0 or 1, and is a positive integer. We give necessary and sufficient conditions for ... such inequalities to define facets of the set covering polytope associated to a 0,1 constraint matrix A. These conditions are in terms of critical edges and critical cutsets defined in the bipartite incidence graph associated to A, and are very much in the spirit of the work of Balas and Zemel on the set packing problem where similar notions were defined in the intersection graph of A. Furthermore, we give a polynomial characterization of a class of 0,1 facets defined from chorded cycles induced in the bipartite incidence graph. This characterization also yields all the 0,1 liftings of odd-hole inequalities for the simple plant location polytope.
Strengthened clique-family inequalities for the stable set polytope
Operations Research Letters, 2021
The stable set polytope is a fundamental object in combinatorial optimisation. Among the many valid inequalities that are known for it, the clique-family inequalities play an important role. Pêcher and Wagler showed that the clique-family inequalities can be strengthened under certain conditions. We show that they can be strengthened even further, using a surprisingly simple mixed-integer rounding argument.