On relaxations of the max k-cut problem formulations

Journal Article · 09 August 2023 · Operations Research Letters

DOI: https://doi.org/10.1016/j.orl.2023.08.001 · OSTI ID:2424029

Fakhimi, Ramin  ^[1]; Validi, Hamidreza  ^[2]; Hicks, Illya V.  ^[3]; Terlaky, Tamás ^[1]; Zuluaga, Luis F.  ^[1]

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1. Lehigh University, PA (United States)
2. Texas Tech University, TX (United States)
3. Rice University, TX (United States)

Here, 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.

• Research Organization: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF)
• Sponsoring Organization: Defense Advanced Research Projects Agency (DARPA); USDOE Office of Science (SC), Basic Energy Sciences (BES). Scientific User Facilities (SUF)
• Grant/Contract Number: AC05-00OR22725
• OSTI ID: 2424029
• Journal Information: Operations Research Letters, Journal Name: Operations Research Letters Journal Issue: 5 Vol. 51; ISSN 0167-6377
• Publisher: ElsevierCopyright Statement
• Country of Publication: United States
• Language: English