Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs

Eran Halperin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

135 Scopus citations

Abstract

We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a performance ratio of 2 - (1 - o(1)) 2 ln ln Δ/ln Δ for large Δ, which improves the previously known ratio of 2- log Δ+O(1)/Δ obtained by Halldórsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For k-uniform hypergraphs with n vertices, we achieve a ratio of k - (1 - o(1))k ln ln n/ln n for large n, and for k-uniform hypergraphs with maximum degree at most Δ the algorithm achieves a ratio of k - (1 - o(1))k(k-1) ln ln Δ/ln Δ for large Δ. These results considerably improve the previous best ratio of k(1 - c/Δ1/k-1) for bounded degree k-uniform hypergraphs, and k(1 - c/nk-1/k) for general k-uniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldórsson.

Original languageEnglish
Pages (from-to)1608-1623
Number of pages16
JournalSIAM Journal on Computing
Volume31
Issue number5
DOIs
StatePublished - May 2002

Keywords

  • Approximation algorithms
  • Semidefinite programming
  • Vertex cover

Fingerprint

Dive into the research topics of 'Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs'. Together they form a unique fingerprint.

Cite this