Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k satisfiability problem

Abstract

Two algorithms for the k-satisfiability problem are presented and a probabilistic analysis is performed. The analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics. The algorithms assign values to literals appearing in a given instance of k-satisfiability, one at a time, until a solution is found or it is discovered that further assignments cannot lead to finding a solution. One algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise. The other algorithm uses a generalization of the unit-clause rule as a heuristic for selecting the next literal: at each step a literal is chosen randomly from a clause containing the least number of literals. The algorithms run in polynomial time and it is shown that they find a solution to a random instance of k-satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values. It is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values.