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Open AccessArticle

Clustering Improves the Goemans–Williamson Approximation for the Max-Cut Problem

Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
Bioinformatics Group, Department of Computer Science, Universität Leipzig, 04107 Leipzig, Germany
Department of Computer Science, University of California, Santa Cruz, CA 95064, USA
Centro de Investigación en Computación, Instituto Politécnico Nacional, Mexico City 07738, Mexico
Institute for Theoretical Chemistry, University of Vienna, 1090 Wien, Austria
Facultad de Ciencias, Universidad National de Colombia, Sede Bogotá 111321, Colombia
Santa Fe Insitute, Santa Fe, NM 87501, USA
Authors to whom correspondence should be addressed.
Computation 2020, 8(3), 75;
Received: 29 June 2020 / Revised: 11 August 2020 / Accepted: 24 August 2020 / Published: 26 August 2020
MAX-CUT is one of the well-studied NP-hard combinatorial optimization problems. It can be formulated as an Integer Quadratic Programming problem and admits a simple relaxation obtained by replacing the integer “spin” variables xi by unitary vectors vi. The Goemans–Williamson rounding algorithm assigns the solution vectors of the relaxed quadratic program to a corresponding integer spin depending on the sign of the scalar product vi·r with a random vector r. Here, we investigate whether better graph cuts can be obtained by instead using a more sophisticated clustering algorithm. We answer this question affirmatively. Different initializations of k-means and k-medoids clustering produce better cuts for the graph instances of the most well known benchmark for MAX-CUT. In particular, we found a strong correlation of cluster quality and cut weights during the evolution of the clustering algorithms. Finally, since in general the maximal cut weight of a graph is not known beforehand, we derived instance-specific lower bounds for the approximation ratio, which give information of how close a solution is to the global optima for a particular instance. For the graphs in our benchmark, the instance specific lower bounds significantly exceed the Goemans–Williamson guarantee. View Full-Text
Keywords: algorithms; approximation; semidefinite programming; Max-Cut; clustering algorithms; approximation; semidefinite programming; Max-Cut; clustering
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Rodriguez-Fernandez, A.E.; Gonzalez-Torres, B.; Menchaca-Mendez, R.; Stadler, P.F. Clustering Improves the Goemans–Williamson Approximation for the Max-Cut Problem. Computation 2020, 8, 75.

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