Improved approximation of max-cut on graphs of bounded degree

Authors:
Uriel Feige;Marek Karpinski;Michael Langberg
Affiliations:
Department of Computer Science and Applied Mathematics, the Weizmann Institute of Science, Rehovot, Israel;Department of Computer Science, University of Bonn, Bonn, Germany;Department of Computer Science and Applied Mathematics, the Weizmann Institute of Science, Rehovot 76100, Israel
Venue:
Journal of Algorithms
Year:
2002

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Abstract

Let α ≃ 0.87856 denote the best approximation ratio currently known for the Max-Cut problem on general graphs. We consider a semidefinite relaxation of the Max-Cut problem, round it using the random hyperplane rounding technique of Goemans and Williamson [J. ACM 42 (1995) 1115-1145], and then add a local improvement step. We show that for graphs of degree at most Δ, our algorithm achieves an approximation ratio of at least α + ε, where ε 0 is a constant that depends only on Δ. Using computer assisted analysis, we show that for graphs of maximal degree 3 our algorithm obtains an approximation ratio of at least 0.921, and for 3-regular graphs the approximation ratio is at least 0.924. We note that for the semidefinite relaxation of Max-Cut used by Goemans and Williamson the integrality gap is at least 1/0.885, even for 2-regular graphs.