Improved approximation of Max-Cut on graphs of bounded degree

Uriel Feige, Marek Karpinski, Michael Langberg

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)201-219
Number of pages19
JournalJournal of Algorithms
Issue number2
StatePublished - May 2002
Externally publishedYes

Bibliographical note

Funding Information:
We thank Monique Laurent for commenting on an early version of our paper. The first author is the Incumbent of the Joseph and Celia Reskin Career Development Chair. This research was supported in part by a Minerva grant, and by DFG grant 673/4-1, Esprit BR grants 7079, 21726, and EC-US 030, and by the Max-Planck Research Prize.


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