Simple Odd $$\beta $$-Cycle Inequalities for Binary Polynomial Optimization

Abstract

We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd β-cycle inequalities valid for this polytope, showed that these generally have Chvátal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd β-cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chvátal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs.

Simple odd $$\beta $$-cycle inequalities for binary polynomial optimization

“…, N , taking values in {+1, −1}. In the formula (8), N and R are positive constants, and the formula is not equal to a constant for R ∈ {3, . .…”

“…Note that (8) coincides with the energy function (2) in [24], up to multiplication by a positive constant, and renaming of variables so that the first variable is σ 1 rather than σ 0 . To express the energy function (8) in 0/1 variables, we replace σ i with 2x i − 1, for i = 1, . .…”

“…Second, simple odd β-cycle inequalities, together with standard linearization inequalities and flower inequalities with at most two neighbors, provide a perfect formulation of the multilinear polytope for cycle hypergraphs. While the above results were first presented in the preliminary IPCO version of this paper [8], in this paper we also discuss redundancy and computations. In fact, we provide computational evidence showing that our separation algorithm can lead to significant speedups in solving several applications that can be formulated as (1) with a hypergraph that contains β-cycles.…”

The Binary Linearization Complexity of Pseudo-Boolean Functions

“…Perfect formulations are known for the first two considered linearizations when considering every linearization variables separately. Strengthening the joint formulation for multiple linearization variables from C is subject of current research, e.g., by means of studying multilinear polytopes [4,5,6,7,8,9,10,11,19]. For B we cannot hope to identify perfect formulations since this encompasses arbitrary binary sets.…”

Efficient linear reformulations for binary polynomial optimization problems

“…They also derive more general conclusions concerning the resulting polyhedron, and study the computational impact of these inequalities in [14]. More recent papers have uncovered yet more classes of valid inequalities and associated polyhedral results [10,15].…”