Space proof complexity for random 3-CNFs

We investigate the space complexity of refuting 3-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random 3-CNF φ in n variables requires, with high probability, Ω(n) distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation φ requires, with high probability, Ω(n) clauses each of width Ω(n) to be kept at the same time in memory. This gives a Ω(n) lower bound for the total space needed in Resolution to refute φ. These results are best possible (up to a constant factor) and answer questions about space complexity of 3-CNFs posed in [FLN12, FLM13, BGT14, BG]. The main technical innovation is a variant of Hall’s Lemma. We show that in bipartite graphs G with bipartition (L,R) and left-degree at most 3, L can be covered by certain families of disjoint paths, called VW-matchings, provided that L expands in R by a factor of (2 − ), for < 1 23 . ∗Computer Science Department, University of Toronto, 10 Kings College Road, M5S 3G4 Toronto, Canada, {patrickb, molloy}@cs.toronto.edu. †Computer Science Department, Sapienza University of Rome, via Salaria 113, 00198 Rome, Italy, {bonacina, galesi, huynh, wollan}@di.uniroma1.it. ‡Supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/20072013)/ERC Grant Agreement no. 279558. ar X iv :1 50 3. 01 61 3v 3 [ cs .C C ] 2 A pr 2 01 5