On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems

• Published in: SIAM journal on computing Vol. 30; no. 5; pp. 1462 - 1484
• Main Authors: Bonet, Maria Luisa, Esteban, Juan Luis, Galesi, Nicola, Johannsen, Jan
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2000
• Subjects: Algorithms; Automation; Boolean; Circuits
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/S0097539799352474

An exponential lower bound for the size of tree-like cutting planes refutations of a certain family of conjunctive normal form (CNF) formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and cutting planes. In both cases only superpolynomial separations were known [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467; J. Johannsen, Inform. Process. Lett., 67 (1998), pp. 37--41; P. Clote and A. Setzer, in Proof Complexity and Feasible Arithmetics, Amer. Math. Soc., Providence, RI, 1998, pp. 93--117]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [ Combinatorica, 19 (1999), pp. 403--435] are extended to monotone real circuits. An exponential separation is also proved between tree-like resolution and several refinements of resolution: negative resolution and regular resolution. Actually, this last separation also provides a separation between tree-like resolution and ordered resolution, and thus the corresponding superpolynomial separation of [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [A. Goerdt, Ann. Math. Artificial Intelligence, 6 (1992), pp. 169--184].