... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400-variable 3-SAT problems in about 2 hours on the average. In general, it can solve hard n-variable random 3-SAT problems with search trees of size O(2 n=18:7 ). In addition to justifying these claims, this dissertation describes the most significant achievements of other researchers in this area, and discusses all of the widely known general techniques for speeding up SAT search algorithms. It should be useful to anyone interested in NP-complete problems or combinatorial optimization in general, and it should be particularly useful to researchers in either Artificial Intelligence or Operations Research.

. The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computer-aided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, computer architecture design, and computer network design. Traditional methods treat SAT as a discrete, constrained decision problem. In recent years, many optimization methods, parallel algorithms, and practical techniques have been developed for solving SAT. In this survey, we present a general framework (an algorithm space) that integrates existing SAT algorithms into a unified perspective. We describe sequential and parallel SAT algorithms including variable splitting, resolution, local search, global optimization, mathematical programming, and practical SAT algorithms. We give performance evaluation of some existing SAT algorithms. Finally, we provide a set of practical applications of the sat...

Abstract. We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more. 1

We prove the worst-case upper bound 1:5045 n for the time complexity of 3-SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2- and 3-clauses, called "blocked clauses", generalizing the extension rule of "Extended Resolution." Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3-clause-sets and can be applied also to arbitrary trees. Keywords: 3-SAT, worst-case upper bounds, analysis of algorithms, Extended Resolution, blocked clauses, generalized autarkness. 1 Introduction In this paper we study the exponential part of time complexity for 3-SAT decision and prove the worst-case upper bound 1:5044:: n for n the number of variables in the input formula, using new algorithmic methods as well as new methods for the analysis. These methods also deepen the already existing approaches in a systematically manner. The following results...

A new algorithm for testing satisfiability of propositional formulas in conjunctive normal form (CNF) is described. It applies reasoning in the form of certain resolution operations, and identification of equivalent literals. Resolution produces growth in the size of the formula, but within a global quadratic bound; most previous methods avoid operations that produce any growth, and generally do not identify equivalent literals. Computational experience indicates that the method does substantially less "guessing" than previously reported algorithms, while keeping a polynomial time bound on the work done between guesses. Experiments indicate that, for larger problems, the time investment in reasoning returns a profit in reduced searching, and the profit increases with increasing problem size. Experimental data compares six branching strategies of the proposed algorithm on a variety of problems, including several Dimacs benchmarks. These branching strategies were shown to perform differently with statistical signi cance. A new scheme based on Johnson's maximum satisfiability approximation algorithm was found to be the best overall. Both satisfiable and unsatifi able random 3-CNF formulas with 50-283 variables and 4.27 ratio of clauses to variables have been tested; this class is generally acknowledged to be relatively "hard" and

We consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3list -coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3-SAT is equivalent to (2, 3)-CSP while the other problems above are special cases of (3, 2)-CSP. We give a fast algorithm for (3, 2)- CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of Davis-Putnam-style backtracking with more sophisticated matching and network flow based ideas. 1 Introduction There has recently been growing interest in analysis of superpolynomial-time algorithms, including algorithms for NP-hard problems such as satisfiability or graph coloring. This interest has multiple causes: . Many important applications can be modeled with these problems, and with the increased speed of modern computers, solved effectively; for instance it is now routine to solve hard 500-variable satisfia...

The (unweighted) Maximum Satisfiability problem (MaxSat) is: given a boolean formula in conjunctive normal form, find a truth assignment that satisfies the most number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved in time O(|F | 1.3803 K ), where |F | is the length of a formula F in conjunctive normal form and K is the number of clauses in F . We also prove the time bounds O(|F |1.3995 k ), where k is the maximum number of satisfiable clauses, and O(1.1279 |F | ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K ). # An extended abstract of this paper was presented at the 26th International Colloquium on Automata, Languages, and Programming (ICALP'99), LNCS 1644, Springer-Verlag, pages 575--584, held in Prague, Czech Republic, July 11-15, 1999. + Supported by a Feodor Lynen fellowship (1998) of the Alexander von HumboldtStiftung, Bonn, and the Center for Discrete Ma...

We show how DNA molecules and stan- dard lab techniques may be used to create a nondeterministic Turing machine. This is the first scheme that shows how to make a universal computer with DNA. We claim that both our scheme and previous ones will work but they probably cannot be scaled up to be of practical computational importance. In vivo,

We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of AT-free graphs the running time of our algorithms can be reduced to O(1.4142 n).

In several applicative fields, the generic system or structure to be designed can be encoded as a CNF formula, which should have a well-defined satisfiability property (either to be satisfiable or to be unsatisfiable). Within a complete solution frame- work, we develop an heuristic procedure which is able, for unsatisfiable instances, to locate a set of clauses causing unsatisfiability. That corresponds to the part of the system that we respectively need to re-design or to keep when we respectively want a satisfiable or unsatisfiable formula. Such procedure can guarantee to find an unsatisfiable subformula, and is aimed to find an approximation of a minimum unsatisfiable subformula. Successful results on both real life data collecting problems and Dimacs problems are presented.