Random CNFs require spacious Polynomial Calculus refutations

We study the space required by Polynomial Calculus refutations of random k-CNFs. We are interested in how many monomials one needs to keep in memory to carry on a refutation. More precisely we show that for k ≥ 4 a refutation of a random k-CNF of ∆n clauses and n variables requires monomial space Ω(n∆− 1+ k−3− ) with high probability. For constant ∆ we prove that monomial space complexity is Θ(n) with high probability. This solves a problem left open in Alekhnovich et al. (STOC, 2000) and in Ben-Sasson, Galesi (FOCS, 2001; Random Struct. Algorithms, 2003). We study the twofold matching game: it is a prover-delayer game on a bipartite graph in which the prover wants to show that the left side has no pair of disjoint matching sets on the right side. The prover has a bounded amount of memory. We show that any delayer’s winning strategy against such prover is also a strategy to satisfy all equations in a bounded memory polynomial calculus refutation. We show that a random k-CNF with k ≥ 4 has large enough expansion with high probability. This allows lower bounds on the memory of a winning prover in the corresponding twofold matching game. A lower bound on the monomial space required to refute the formula follows. We claim without proof that our result also applies to pigeonhole principles on bipartite graphs. ∗Dipartimento di Informatica, Sapienza Università di Roma. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 137 (2009)