Average time analysis of simplified Davis-Putnam procedures☆

The computational decision problem CPA [1, 2, 3, 4] is a variant of the probabilitistic satisfiability problem PSAT [5,11,15]. In this paper we investigate the behavior of an algorithm which decides CPA applied to the still NP-complete subproblem 2CPA, instances have at most two literals per clause. We locate, as it is done for some satisfiability problems [10,12, 13, 14,17], a critical value for the ratio a = m/n, m is the number of binary clauses present in the instance and n is the number of events. point divides “almost all coherent” instances from “almost all not coherent”: the most difficult instances lies near this point.

This is a written record of a survey talk on the topic of typical case complexity of Satisfiability algorithms which was presented at the LICS workshop on this subject in June, 2003, Ottawa, Canada.

We present a history of results related to the threshold phenomena which arise in the study of random conjunctive normal form (CNF) formulas. In a companion paper (D. Achlioptas, Theoret. Comput. Sci., this volume) the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal, intuitive manner.

Many efficient algorithms have been developed for satisfiability testing. They complement rather than exclude each other by being effective for particular problem instances. In this research, we give a Multi-SAT algorithm for the SAT problem. The Multi-SAT algorithm integrates several efficient SAT algorithms. It makes use of different algorithmic niches for satisfiability testing. Based on cost-effective cluster computing, Multi-SAT can perform simultaneous satisfiability testing, using several “stones” to shoot one “bird”. The software architecture for Multi-SAT has been designed. A number of software tools have been developed. This software tool kit can support efficient satisfiability testing with uncertain problem structure, facilitating multiple tracking of an algorithm structure, and allow a detailed study of the entire problem spectrum. It provides a cost-effective multi-tool kit for practical satisfiability testing.

Experiments on solvingr-SAT random formulae have provided evidence of a satisfiability threshold phenomenon with respect to the ratio of the number of clauses to the number of variables of formulae. Presently, only the threshold of 2-SAT formulae has been proved to exist and has been computed to be equal to 1. For 3-SAT formulae and more generally forr-SAT formulae, lower and upper bounds of the threshold have been established. The best established bounds concern 3-SAT. For an observed threshold of about 4.25, the best lower bound is 3.003 and the best upper bound 4.76. In this paper we establish a general upper bound of the threshold forr-SAT formulae giving a value for 3-SAT of 4.64, significantly improving the previous best upper bound. For this we have defined a more restrictive structure than a satisfying truth assignment for characterizing the satisfiability of a SAT formula which we have called negatively prime solution (NPS). By merely applying the first moment method to negatively prime solutions of a randomr-SAT formula we obtain our bound.

Domain independent general purpose problem solving techniques are desirable from the standpoints of software engineering and human computer interaction. They employ declarative and modular knowledge representations and present a constant homogeneous interface to the user, untainted by the peculiarities of the specific domain of interest. Unfortunately, this very insulation from domain details often precludes effective problem solving behavior. General approaches have proven successful in complex real-world situations only after a tedious cycle of manual experimentation and modification. Machine learning offers the prospect of automating this adaptation cycle, reducing the burden of domain specific tuning and reconciling the conflicting needs of generality and efficacy. A principal impediment to adaptive techniques is the utility problem: even if the acquired information is accurate and is helpful in isolated cases, it may degrade overall problem solving performance under difficult to predict circumstances. We develop a formal characterization of the utility problem and introduce COMPOSER, a statistically rigorous learning approach which avoids the utility problem. COMPOSER has been successfully applied to learning heuristics for planning and scheduling systems. This article includes theoretical results and an extensive empirical evaluation. The approach is shown to outperform significantly several other leading approaches to the utility problem.