Resolution complexity of independent sets in random graphs

Abstract

We consider the problem of providing a resolution proof of the statement that a given graph with n vertices and /spl Delta/n edges does not contain an independent set of size k. For randomly chosen graphs with constant /spl Delta/, we show that such proofs almost surely require size exponential in n. Further, for /spl Delta/=o(n/sup 1/5/) and any k/spl les/n/5, we show that these proofs almost surely require size 2(n/sup /spl delta//) for some global constant /spl delta/>0, even though the largest independent set in graphs with /spl Delta//spl ap/n/sup 1/5/ is much smaller than n/5. Our result shows that almost all instances of the independent set problem are hard for resolution. It also provides a lower bound on the running time of a certain class of search algorithms for finding a largest independent set in a given graph.