Binary positive semidefinite matrices and associated integer polytopes (original) (raw)

Polytopes of Minimum Positive Semidefinite Rank

Discrete & Computational Geometry, 2013

The positive semidefinite (psd) rank of a polytope is the smallest k for which the cone of k × k real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization in dimensions two and three.

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 skeleton of the dual cut polytope

Contemporary Mathematics, 1994

The cut polytope is the ? n 2 -dimensional convex polytope generated by all cuts of the complete graph on n nodes. One of the applications of the cut polytope, the polyhedral approach to the maximum cut problem, leads to the study of its facets which are known only up to n = 7 where they number 116764. For n 7, we describe the skeleton of the dual of the cut polytope, in particular, we give its adjacencies relations and diameter. We also give similar results for a relative of the cut polytope, the cut cone, and new results on the size of the facets of the cut polytope.

Some efficiently solvable problems over integer partition polytopes

Discrete Applied Mathematics, 2015

The integer partition polytope P n is the convex hull of all integer partitions of n. We provide a novel extended formulation of P n , and use it to show that the extremality, adjacency, and separation problems over P n can be solved by linear programming without the ellipsoid method.

On a positive semidefinite relaxation of the cut polytope

Linear Algebra and its Applications, 1995

General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Some characteristics of the simple Boolean quadric polytope extension

arXiv: Combinatorics, 2016

Following the seminal work of Padberg on the Boolean quadric polytope BQPBQPBQP and its LP relaxation BQPLPBQP_{LP}BQPLP, we consider a natural extension: SATPSATPSATP and SATPLPSATP_{LP}SATPLP polytopes, with BQPLPBQP_{LP}BQPLP being projection of the SATPLPSATP_{LP}SATPLP face (and BQPBQPBQP - projection of the SATPSATPSATP face). We consider the problem of integer recognition: determine whether the maximum of a linear objective function is achieved at an integral vertex of a polytope. Various special instances of 3-SAT problem like NAE-3-SAT, 1-in-3-SAT, weighted MAX-3-SAT, and others can be solved by integer recognition over SATPLPSATP_{LP}SATPLP. We study the properties of SATPSATPSATP integral vertices. Like BQPLPBQP_{LP}BQPLP, polytope SATPLPSATP_{LP}SATPLP has the Trubin-property being quasi-integral (1-skeleton of SATPSATPSATP is a subset of 1-skeleton of SATPLPSATP_{LP}SATPLP). However, unlike BQPBQPBQP, not all vertices of SATPSATPSATP are pairwise adjacent, the diameter of SATPSATPSATP equals 2, and the clique number of 1-skeleton is superpolynomial in dimension. It is known that the fracti...

On Polytopes with Linear Rank with respect to Generalizations of the Split Closure

2021

In this paper we study the rank of polytopes contained in the 0-1 cube with respect to t-branch split cuts and t-dimensional lattice cuts for a xed positive integer t . These inequalities are the same as split cuts when t = 1 and generalize split cuts when t > 1. For polytopes contained in the n-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most n, and this bound is tight as Cornuéjols and Li gave an example with split rank n. All known examples with high split rank – i.e., at least cn for some positive constant c < 1 – are de ned by exponentially many (as a function of n) linear inequalities. For any xed integer t > 0, we give a family of polytopes contained in [0, 1]n for su ciently large n such that each polytope has empty integer hull, is de ned by O(n) inequalities, and has rank Ω(n) with respect to t-dimensional lattice cuts. Therefore the split rank of these polytopes is Ω(n). It was shown earlier that there exist generalized branch-an...

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.

The Integer Hull of a Convex Rational Polytope

Discrete and Computational Geometry, 2004

Given A ∈ Z m×n and b ∈ Z m , we consider the integer program max{c x|Ax = b; x ∈ N n } and provide an equivalent and explicit linear program max{ c q|Mq = r; q ≥ 0}, where M, r, c are easily obtained from A, b, c with no calculation. We also provide an explicit algebraic characterization of the integer hull of the convex polytope P = {x ∈ R n |Ax = b; x ≥ 0}. All strong valid inequalities can be obtained from the generators of a convex cone whose definition is explicit in terms of M.

On the Graph Bisection Cut Polytope

SIAM Journal on Discrete Mathematics, 2008

Given a graph G = (V, E) with node weights ϕ v ∈ N ∪ {0}, v ∈ V , and some number F ∈ N∪{0}, the convex hull of the incidence vectors of all cuts δ(S), S ⊆ V with ϕ(S) ≤ F and ϕ(V \ S) ≤ F is called the bisection cut polytope. We study the facial structure of this polytope which shows up in many graph partitioning problems with applications in VLSI-design or frequency assignment. We give necessary and in some cases sufficient conditions for the knapsack tree inequalities introduced in [9] to be facet-defining. We extend these inequalities to a richer class by exploiting that each cut intersects each cycle in an even number of edges. Finally, we present a new class of inequalities that are based on non-connected substructures yielding non-linear right-hand sides. We show that the supporting hyperplanes of the convex envelope of this non-linear function correspond to the faces of the so-called cluster weight polytope, for which we give a complete description under certain conditions.