Graph Expansion, Tseitin Formulas and Resolution Proofs for CSP

We study the resolution complexity of Tseitin formulas over arbitrary rings in terms of combinatorial properties of graphs. We give some evidence that the 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 bound on the size of resolution proofs to the constraint satisfaction problem under an arbitrary finite alphabet. For Tseitin formulas under the alphabet of cardinality d we prove stronger lower bound de(G)−k on the tree-like resolution complexity, where e(G) is the graph expansion that is equal to the minimal cut such that sizes of its parts differ in at most 2 times and k is an upper bound on the degree of the graph. 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)·log 3 2 |V | for some subgraph H of G.