Satisfiability, Branch-Width and Tseitin Tautologies

• Authors: Michael Alekhnovich;Alexander A. Razborov
• Venue: FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
• Year: 2002

For a CNF_\tau, let wb(\tau ) be the bronch-width of its underlying hypergraph. In this paper we design an algorithm for solving SAT in time n^{0(1)} 2^0 (wb(\tau )). This in particular implies a polynomial algorithm for testing satisfiability on instances with tree-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. We show that instead of the exhaustive enumerotion of all provable clauses, one can do a better search based on the Robertson-Seymour algorithm forapproximating the bronch-width of a graph. We call the resulting procedure Bronch- Width Based Automated Theorem Prover (BWBATP). As opposed to WBATP, it always produces regular refutations. Perhaps more importantly, the running time of our algorithm is boundedin terms of a clean combinatorial charocteristic that can be efficiently approximated, and that the algorithm also produces, within the same time, a satisfying assignment if \tauhappens 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 \tau is \left| \tau\right|^{0(1)}\cdot 2^{0(w(\tau 1 - 0)} as opposed to the obvious bound n^{0(1)} 2^0 (wb(\tau ))\left| \tau\right|^{0(1)}\cdot 2^{0(w(\tau 1 - 0)} provided by WBATP.This in particular implies that Resolution is automatizable on those Tseitin tautologies for which we know the relation w(\tau 1 - 0) \leqslant 0(\log S(\tau )). We identify one such subclass and prove partial results toward establishing this relation for larger classes of graphs.