Space proof complexity for random $3$-CNFs via a $(2-\epsilon)$-Hall's Theorem

We investigate the space complexity of refuting 3-CNFs in Resolution and algebraic systems. No lower bound for refuting any family of 3-CNFs was previously known for the total space in resolution or for the monomial space in algebraic systems. Using the framework of [10], we prove that every Polynomial Calculus with Resolution refutation of a random 3-CNF φ in n variables requires, with high probability, Ω(n/ log 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/ log n) clauses each of width Ω(n/ log n) to be kept at the same time inmemory. This gives aΩ(n/ log n) lower bound for the total space needed in Resolution to refute φ. The results answer questions about space complexity of 3-CNFs posed in [16, 15, 11, 10]. The main technical innovation is a variant ofHall’s theorem. 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 (2, 4)-matchings, provided that L expands in R by a factor of (2 − ǫ), for ǫ < 1 23 .