Graph Expansion, Tseitin Formulas and Resolution Proofs for CSP

Abstract

We study the resolution complexity of Tseitin formulas over arbitrary rings in terms of combinatorial properties of graphs. We give some evidence that an expansion of a graph is a good characterization of the resolution complexity of Tseitin formulas. We extend the method of Ben-Sasson and Wigderson of proving lower bounds for the size of resolution proofs to constraint satisfaction problems under an arbitrary finite alphabet. For Tseitin formulas under the alphabet of cardinality d we provide a lower bound d ^{e(G) − k} for tree-like resolution complexity that is stronger than the one that can be obtained by the Ben-Sasson and Wigderson method. Here k is an upper bound on the degree of the graph and e(G) is the graph expansion that is equal to the minimal cut such that none of its parts is more than twice bigger than the other. We give a formal argument why a large graph expansion is necessary for lower bounds. Let G = 〈V,E 〉 be the dependency graph of the CSP, vertices of G correspond to constraints; two constraints are connected by an edge for every common variable. We prove that the tree-like resolution complexity of the CSP is at most $d^{e(H) \cdot \log_{\frac{3}{2}} |V|}$ for some subgraph H of G.