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Open AccessArticle

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

1
Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
2
Bioinformatics Group, Department of Computer Science, Universität Leipzig, 04107 Leipzig, Germany
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Department of Computer Science, University of California, Santa Cruz, CA 95064, USA
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Centro de Investigación en Computación, Instituto Politécnico Nacional, Mexico City 07738, Mexico
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Institute for Theoretical Chemistry, University of Vienna, 1090 Wien, Austria
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Facultad de Ciencias, Universidad National de Colombia, Sede Bogotá 111321, Colombia
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Santa Fe Insitute, Santa Fe, NM 87501, USA
*
Authors to whom correspondence should be addressed.
Computation 2020, 8(3), 75; https://doi.org/10.3390/computation8030075
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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