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 \(n\times n\) grid graph has size at least \(2^{\varOmega (n)}\). Then using the Excluded Grid Theorem by Robertson and Seymour we show that for arbitrary graph G(V, E) any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on G has size at least \(2^{\varOmega (\mathrm {tw}(G)^\delta )}\) for all \(\delta <1/36\), where \(\mathrm {tw}(G)\) is the treewidth of G (for planar graphs and some other classes of graphs the statement holds for \(\delta =1\)). We also show an upper bound of \(O(|E| 2^{\mathrm {pw}(G)})\), where \(\mathrm {pw}(G)\) is the pathwidth of G.