Compact Linearization for Binary Quadratic Problems subject to Assignment Constraints

TL;DR

This paper introduces new necessary and sufficient conditions for a compact linearization of binary quadratic problems with assignment constraints, resolving previous inconsistencies and providing efficient algorithms for minimal linearization.

Contribution

It establishes a complete characterization of linearization conditions, proposes a MILP for minimal linearization size, and offers a polynomial-time algorithm for special cases.

Findings

New conditions ensure consistent linearization.

MILP computes minimal linearization size.

Polynomial algorithm is exact for non-overlapping support cases.

Abstract

We prove new necessary and sufficient conditions to carry out a compact linearization approach for a general class of binary quadratic problems subject to assignment constraints as it has been proposed by Liberti in 2007. The new conditions resolve inconsistencies that can occur when the original method is used. We also present a mixed-integer linear program to compute a minimally-sized linearization. When all the assignment constraints have non-overlapping variable support, this program is shown to have a totally unimodular constraint matrix. Finally, we give a polynomial-time combinatorial algorithm that is exact in this case and can still be used as a heuristic otherwise.