On Tseitin Formulas, Read-Once Branching Programs and Treewidth

Abstract

We show that any nondeterministic read-once branching program that decides a satisfiable Tseitin formula based on an n × n grid graph has size at least 2^Ω(n). Then using the Excluded Grid Theorem by Robertson and Seymour we show that for an arbitrary graph G(V, E) any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on G has size at least $2^{\Omega (\text {tw}(G)^{\delta })}$ for all δ < 1/36, where tw(G) is the treewidth of G (for planar graphs and some other classes of graphs the statement holds for δ = 1). We apply the mentioned results to the analysis of the complexity of derivations in the proof system OBDD(∧,reordering) and show that any OBDD(∧,reordering)-refutation of an unsatisfiable Tseitin formula based on a graph G has size at least $2^{\Omega (\text {tw}(G)^{\delta })}$. We also show an upper bound O(|E|2^pw(G)) on the size of OBDD representations of a satisfiable Tseitin formula based on G and an upper bound $O(|E||V| 2^{\text {pw}(G)}+|\text {TS}_{G,c}|^{2})$ on the size of OBDD(∧)-refutation of an unsatisifable Tseitin formula TS_{G, c}, where pw(G) is the pathwidth of G.