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 $2-(1-o(1))\frac{2 \ln \ln \Delta}{\ln \Delta}$ for large $\Delta$, which improves the previously known ratio of $2-\frac{\log \Delta + O(1)}{\Delta}$ obtained by Halld{orsson 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 $k-(1-o(1))\frac{k\ln \ln n}{\ln n}$ for large n, and for k-uniform hypergraphs with maximum degree at most $\Delta$ the algorithm achieves a ratio of $k-(1-o(1))\frac{k(k-1)\ln \ln \Delta}{\ln \Delta}$ for large $\Delta$. These results considerably improve the previous best ratio of $k(1-c/\Delta^\frac{1}{k-1})$ for bounded degree k-uniform hypergraphs, and $k(1-c/n^\frac{k-1}{k})$ 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.