Tight Lower Bounds on the Resolution Complexity of Perfect Matching Principles

Issue title: Discrete Mathematics (RuFiDiM 14)

Guest editors: J. Karhumäki, V. Mazalov and Yu. Matiyasevich

Article type: Research Article

Authors: Itsykson, Dmitry^{a; *; †} | Oparin, Vsevolod^b | Slabodkin, Mikhail^c | Sokolov, Dmitry^d

Affiliations: [a] Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, St.Petersburg, 191023, Russia. [email protected] | [b] St. Petersburg Academic University, 8 Khlopina, St.Petersburg, 194021, Russia. [email protected] | [c] St. Petersburg Academic University, 8 Khlopina, St.Petersburg, 194021, Russia. [email protected] | [d] Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, St.Petersburg, 191023, Russia. [email protected]

Correspondence: [†] Address for correspondence: Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, St.Petersburg, 191023, Russia

Note: [*] The research is partially supported by the RFBR grant 14-01-00545, by the President’s grant MK-2813.2014.1 and by the Government of the Russia (grant 14.Z50.31.0030).

Abstract: The resolution complexity of the perfect matching principle was studied by Razborov [1], who developed a technique for proving its lower bounds for dense graphs. We construct a constant degree bipartite graph Gn such that the resolution complexity of the perfect matching principle for Gn is 2Ω(n) where n is the number of vertices in Gn. This lower bound is tight up to some polynomial. Our result implies the 2Ω(n) lower bounds for the complete graph K2n + 1 and the complete bipartite graph Kn,O(n) that improves the lower bounds following from [1]. We show that for every graph G with n vertices that has no perfect matching there exists a resolution refutation of perfect matching principle for G of size O(n22n). Thus our lower bounds match upper bounds up to a multiplicative constant in the exponent. Our results also imply the well-known exponential lower bounds on the resolution complexity of the pigeonhole principle, the functional pigeonhole principle and the pigeonhole principle over a graph. We also prove the following corollary. For every natural number d, for every n large enough, for every function h : {1, 2, . . . , n} → {1, 2, . . . , d}, we construct a graph with n vertices that has the following properties. There exists a constant D such that the degree of the i-th vertex is at least h(i) and at most D, and it is impossible to make all degrees equal to h(i) by removing the graph’s edges. Moreover, any proof of this statement in the resolution proof system has size 2Ω(n). This result implies well-known exponential lower bounds on the Tseitin formulas as well as new results: for example, the same property of a complete graph. Preliminary version of this paper appeared in proceedings of CSR-2015 [2].

Keywords: proof complexity, expander, perfect matching, resolution, width

DOI: 10.3233/FI-2016-1358

Journal: Fundamenta Informaticae, vol. 145, no. 3, pp. 229-242, 2016