Some Characteristics of the Simple Boolean Quadric Polytope Extension

Following the seminal work of Padberg on the Boolean quadric polytope BQP and its LP relaxation BQPLP , we consider a natural extension: SATP and SATPLP polytopes, with BQPLP being projection of the SATPLP face (and BQP projection of the SATP 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 SATPLP . We study the properties of SATP integral vertices. Like BQPLP , polytope SATPLP has the Trubin-property being quasi-integral (1-skeleton of SATP is a subset of 1-skeleton of SATPLP ). However, unlike BQP , not all vertices of SATP are pairwise adjacent, the diameter of SATP equals 2, and the clique number of 1-skeleton is superpolynomial in dimension. It is known that the fractional vertices of BQPLP are halfintegral (0, 1 or 1/2 valued). We show that the denominators of SATPLP fractional vertices can take any integer value. Finally, we describe polynomially solvable subproblems of integer recognition over SATPLP with constrained objective functions. Based on that, we solve some cases of edge constrained bipartite graph coloring. 1. Boolean quadric polytope and its relaxations We consider the well-known Boolean quadric polytope BQP (n) [13], satisfying the constraints xi + xj − xi,j ≤ 1, (1) xi,j ≤ xi, (2) xi,j ≤ xj , (3) xi,j ≥ 0, (4) xi, xi,j ∈ {0, 1}, (5) for all i, j : 1 ≤ i < j ≤ n. Polytope BQP (n) is constructed from the NP-hard problem of unconstrained Boolean quadratic programming: Q(x) = xQx→ max, where vector x ∈ {0, 1}n, and Q is an upper triangular matrix, by introducing new variables xi,j = xixj . Boolean quadric polytope arises in many fields of mathematics and physics. Sometime it is called the correlation polytope, since its members can be interpreted as joint correlations of events in some probability space. Also within the quantum mechanics Boolean quadric polytope is connected with the representability problem for density matrices of order 2 that render physical properties of a system of particles [6]. Besides, BQP (n) is in one-to-one correspondence via the