On relaxations of the max $k$-cut problem formulations

TL;DR

This paper compares different continuous relaxations of the max k-cut problem, showing that a binary quadratic formulation's relaxation is stronger than some linear formulations and comparable to semidefinite relaxations, with empirical validation.

Contribution

It demonstrates the relative strength of various relaxations for max k-cut and provides empirical evidence supporting the effectiveness of quadratic optimization solvers.

Findings

Binary quadratic relaxation is stronger than linear relaxations.

Quadratic relaxation is at least as strong as semidefinite relaxation.

State-of-the-art solvers effectively handle the relaxations.

Abstract

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max $k$-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max $k$-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max $k$-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.