Finding large independent sets in graphs and hypergraphs

A basic problem in graphs and hypergraphs is that of finding a large independent set-one of guaranteed size. Understanding the parallel complexity of this and related independent set problems on hypergraphs is a fundamental open issue in parallel computation. Caro and Tuza [J. Graph Theory, 15 (1991), pp. 99-107] have shown a certain lower bound alpha(k)(H) on the size of a maximum independent set in a given k-uniform hypergraph H and have also presented an efficient sequential algorithm to find an independent set of size alpha k(H). They also show that alpha(k)(H) is the size of the maximum independent set for various hypergraph families. Here, we show that an RNC algorithm due to Beame and Luby [in Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 1990, pp. 212-218] finds an independent set of expected size alpha(k)(H) and also derandomizes it for certain special cases. (An intriguing conjecture of Beame and Luby implies that understanding this algorithm better may yield an RNC algorithm to find a maximal independent set in hypergraphs, which is among the outstanding open questions in parallel computation.) We also present lower bounds on independent set size for nonuniform hypergraphs using this algorithm. For graphs, we get an NC algorithm to find independent sets of size essentially that guaranteed by the general (degree-sequence based) version of Turan's theorem.