A lower bound for tree resolution

Abstract

In this article, highly expanding degree-3 bipartite graphs are generated randomly. Every graph gives rise to a contradictory set of clauses, and these particular graphs provide us with a highly interconnected set of 3SAT clauses. If n is the number of nodes in each side of the graph, then there are 3n variables and 8n clauses. We use this set to prove a lower bound for tree resolution. The lower bound obtained is 2^{($\text{2}\text{3}$λn}) where λ≈0.3166. Letter N = 3n be the number of variables, this bound is ≈ 2^.070355N. This is contrasted with the best-known upper bound for 3SAT, from the algorithm in Monien and Speckenmeyer (1985), which is ≈ 2^.6943N where again N is the number of variables. Exponential lower bounds have been proved for stronger forms of resolution, but with significantly smaller constants.