The Bipartite Boolean Quadric Polytope with Multiple-Choice Constraints

TL;DR

This paper studies a specialized bipartite boolean quadric polytope with multiple-choice constraints, revealing its properties, facet structures, symmetries, and effective algorithms, leading to improved solutions for complex pooling problems.

Contribution

It extends the bipartite boolean quadric polytope framework by analyzing multiple-choice constraints, introducing new facet classes, symmetries, and polynomial-time separation algorithms.

Findings

Polytope inherits many properties from BQP.

New facet classes and symmetries are identified.

Cutting plane methods significantly reduce solution times.

Abstract

We consider the bipartite boolean quadric polytope (BQP) with multiple-choice constraints and analyse its combinatorial properties. The well-studied BQP is defined as the convex hull of all quadric incidence vectors over a bipartite graph. In this work, we study the case where there is a partition on one of the two bipartite node sets such that at most one node per subset of the partition can be chosen. This polytope arises, for instance, in pooling problems with fixed proportions of the inputs at each pool. We show that it inherits many characteristics from BQP, among them a wide range of facet classes and operations which are facet preserving. Moreover, we characterize various cases in which the polytope is completely described via the relaxation-linearization inequalities. The special structure induced by the additional multiple-choice constraints also allows for new facet-preserving symmetries as well as lifting operations. Furthermore, it leads to several novel facet classes as well as extensions of these via lifting. We additionally give computationally tractable exact separation algorithms, most of which run in polynomial time. Finally, we demonstrate the strength of both the inherited and the new facet classes in computational experiments on random as well as real-world problem instances. It turns out that in many cases we can close the optimality gap almost completely via cutting planes alone, and, consequently, solution times can be reduced significantly.