Resolution complexity of independent sets in random graphs

We consider the problem of providing a resolution proof of the statement that a given graph with n vertices and Δn edges does not contain an independent set of size k. For randomly chosen graphs with constant Δ, we show that such proofs almost surely require size exponential in n. Further, for Δ=o(n^1/5) and any k⩽n/5, we show that these proofs almost surely require size 2(n^δ) for some global constant δ>0, even though the largest independent set in graphs with Δ≈n^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