Space complexity of random formulae in resolution
Eli Ben‐Sasson, Nicola Galesi · Random Structures and Algorithms · 2003
Abstract We study the space complexity of refuting unsatisfiable randomk‐CNFs in the Resolution proof system. We prove that for Δ ≥ 1 and any ϵ > 0, with high probability a randomk‐CNF overnvariables and Δnclauses requires resolution clause space of Ω(n/Δ1+ϵ). For constant Δ, 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 Δ ≫n. This bound is nearly tight. Specifically, we show that with high probability, a random 3‐CNF with Δnclauses requires treelike refutation size of exp(Ω(n/Δ1+ϵ)), for any ϵ > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2‐player Matching Game on bipartite graphsGto prove that there are no perfect matchings inG. (2) We reduce lower bounds for the clause space of a formulaFin Resolution to lower bounds for the complexity of the game played on the bipartite graphG(F) associated withF. (3) We prove that the complexity of the game is large wheneverGis an expander graph. Finally, a simple probabilistic analysis shows that for a random formulaF, with high probabilityG(F) is an expander. We also extend our result to the case ofG‐PHP, a generalization of the Pigeonhole principle based on bipartite graphsG. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 92–109, 2003