On good algorithms for determining unsatisfiability of propositional formulas

Abstract

This paper is concerned with an algorithm that provides short certificates of unsatisfiability with high probability when an input I is a random instance of 3-SAT. Rather than build a refutation DAG, the algorithm finds bounds on n I (true) , the number of variables that must be set to true, and n I (false) , the number that must be set to false, if all clauses of I are to be satisfied. If the sum n I (true)+n I (false) is greater than the number of variables in I then I must be unsatisfiable. Bounds on n I (true) and n I (false) may be found by throwing out all clauses except those having only negative or only positive literals and finding n I +(true) for the remaining positive clause set I + and n I − (false) for the remaining negative clause set I − . These questions can alternatively be stated as 3-hitting set problems on I + and I − separately. It is shown that a good enough approximation algorithm for 3-hitting set can determine useful bounds on n I (true) and n I (false) (high probability of success for large enough constant ratio of clauses to variables). Although a good enough algorithm seems evasive, one that comes fairly close is proposed and analyzed.