par Bennett, Patrick;Bonacina, Ilario;Galesi, Nicola;Huynh, Tony 
;Molloy, Mike;Wollan, Paul
Référence Information and computation, 255, page (165-176)
Publication Publié, 2017-08
;Molloy, Mike;Wollan, PaulRéférence Information and computation, 255, page (165-176)
Publication Publié, 2017-08
Article révisé par les pairs
| Résumé : | 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 of 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(n2) 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. The main technical innovation is a variant of Hall's Lemma. We show that in bipartite graphs 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−ϵ), for ϵ<1/5. |
