Space complexity of random formulae in resolution

We study the space complexity of refuting unsatisfiable random k-CNFs in the Resolution proof system. We prove that for Delta greater than or equal to 1 and any epsilon > 0, with high probability a random k-CNF over n variables and Deltan clauses requires resolution clause space of Omega(n/Delta(1+epsilon)). For constant A, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3-CNFs with clause density Delta much greater than rootn. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with Deltan clauses requires treelike refutation size of exp(Omega(n/Delta(1+epsilon))), for any epsilon > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2-player Matching Game on bipartite graphs G to prove that there are no perfect matchings in G. (2) We reduce lower bounds for the clause space of a formula F in Resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. (3) We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the Pigeonhole principle based on bipartite graphs G. (C) 2003 Wiley Periodicals, Inc