Satisfiability, Branch-Width and Tseitin tautologies

Abstract

For a CNF τ, let w _b (τ) be the branch-width of its underlying hypergraph, that is the smallest w for which the clauses of τ can be arranged in the form of leaves of a rooted binary tree in such a way that for every vertex its descendants and non-descendants have at most w variables in common. In this paper we design an algorithm for solving SAT in time ${n^{O(1)}2^{O(w_b(\tau))}}$. This in particular implies a polynomial algorithm for testing satisfiability on instances with branch-width O(log n). Our algorithm is a modification of the width based automated theorem prover (WBATP) which is a popular (at least on the theoretical level) heuristic for finding resolution refutations of unsatisfiable CNFs, and we call it Branch-Width Based Automated Theorem Prover (BWBATP). As opposed to WBATP, our algorithm always produces regular refutations. Perhaps more importantly, its running time is bounded in terms of a clean combinatorial characteristic that can be efficiently approximated, and that the algorithm also produces, within the same time, a satisfying assignment if τ happens to be satisfiable.

In the second part of the paper we investigate the behavior of BWBATP on the well-studied class of Tseitin tautologies. We argue that in this case BWBATP is better than WBATP. Namely, we show that its running time on any Tseitin tautology τ is ${|\tau|^{O(1)} \cdot 2^{O(w(\tau\vdash\emptyset))}}$ , as opposed to the obvious bound ${n^{O(w(\tau\vdash\emptyset))}}$ provided by WBATP.

This in particular implies that resolution is automatizable on those Tseitin tautologies for which we know the relation ${w(\tau\vdash\emptyset)\leq O(\log S(\tau))}$. We identify one such subclass and prove partial results toward establishing this relation for larger classes of graphs.