# Satisfiability, Branch-Width and Tseitin tautologies

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## 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.

## Keywords

SAT provers Automatizability Branch-width## Subject classification

68Q17 68Q25 68T15 03F20## Preview

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