The probabilistic analysis of a greedy satisfiability algorithm
Abstract:
Consider the following simple, greedy Davis-Putnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occurrences of the negation of the literal). Delete these clauses from the formula, and also delete the negation of this literal from any clauses it appears. Repeat. If however unit clauses ever appear, then first repeatedly and in any order set the literals in them to True and delete and shrink clauses accordingly, until no unit clause remains. Also if at any step an empty clause appears, then do not backtrack, but just terminate the algorithm and report failure. A slight modification of this algorithm is probabilistically analyzed in this paper (rigorously). It is proved that for random formulas of n variables and density up to 3.42, it succeeds in producing a satisfying truth assignment with bounded away from zero probability, as n approaches infinity. Therefore the satisfiability threshold is at least 3.42. Previously, Davis-Putnam algorithms of increasing sophistication were analyzed to obtain a lower bound of at most 3.26. Preliminary experiments we performed suggest that the value 3.42 can be considerably increased (up to approximately 3.6) with algorithms like the above, which for the selection of the literal to be set take into account not only its number of occurrences but also the number of occurrences of its negation, but ignore the size of the clauses were the literal appears (apart of the unit-clause, "forced " steps). So, perhaps, for formulas with density up to (a number close to) the satisfiability threshold, a satisfying truth assignment can be found by simple Davis-Putnam algorithms based only on the number of occurrences of the literals.
