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

Ramin Fakhimi; Hamidreza Validi; Illya V. Hicks; Tamás Terlaky; Luis F. Zuluaga

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arXiv · 2308.01361

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

Ramin Fakhimi·Hamidreza Validi·Illya V. Hicks·Tamás Terlaky·Luis F. Zuluaga

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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.

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Ramin Fakhimi, Hamidreza Validi, Illya V. Hicks, Tamás Terlaky, Luis F. Zuluaga. 2023-08-02. On relaxations of the max $k$-cut problem formulations. https://arxiv.org/abs/2308.01361

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