Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1611.01645

Some characteristics of the simple Boolean quadric polytope extension

Andrei Nikolaev
View original source ↗Read source document ↗Export metadata ↓

Abstract

Following the seminal work of Padberg on the Boolean quadric polytope $BQP$ and its LP relaxation $BQP_{LP}$, we consider a natural extension: $SATP$ and $SATP_{LP}$ polytopes, with $BQP_{LP}$ being projection of the $SATP_{LP}$ face (and $BQP$ -- projection of the $SATP$ face). We consider a problem of integer recognition: determine whether a 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 $SATP_{LP}$. We describe all integral vertices of $SATP_{LP}$. Like $BQP_{LP}$, polytope $SATP_{LP}$ has the Trubin-property being quasi-integral (1-skeleton of $SATP$ is a subset of 1-skeleton of $SATP_{LP}$). 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 $BQP_{LP}$ are half-integer (0, 1 or 1/2 valued). We show that the denominators of $SATP_{LP}$ fractional vertices can take any integral value. Finally, we describe polynomially solvable subproblems of integer recognition over $SATP_{LP}$ with constrained objective functions. Based on that, we solve some cases of edge constrained bipartite graph coloring.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andrei Nikolaev. 2017-03-22. Some characteristics of the simple Boolean quadric polytope extension. https://arxiv.org/abs/1611.01645

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗