The linearization problem of a binary quadratic problem and its applications

TL;DR

This paper explores the linearization problem in binary quadratic problems, introduces a new bounding strategy, compares existing bounds, and provides a polynomial-time algorithm for the quadratic shortest path problem.

Contribution

It introduces a linearization-based bounding scheme, compares various bounds, and offers a polynomial-time algorithm for the quadratic shortest path problem.

Findings

Generalized Gilmore-Lawler bounds are linearization-based.

First level reformulation bounds are a type of linearization-based bounds.

A polynomial-time algorithm characterizes linearizable matrices for the quadratic shortest path problem.

Abstract

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem.