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

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 Delta, the algorithm achieves a performance ratio of 2 - (1 - o(1))2 ln ln Delta/ln Delta for large Delta, which improves the previously known ratio of 2 - log Delta+ O (1)/Delta obtained by Halldorsson 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 Delta the algorithm achieves a ratio of k - (1 - o(1))k(k-1) ln ln Delta/ln Delta for large Delta. These results considerably improve the previous best ratio of k(1 - c/Delta 1/k-1) for bounded degree k - uniform hypergraphs, and k(1 - c/n k-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 Halldorsson.