. 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...

. Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case complexity, the threshold phenomenon, known lower bounds for certain classes of algorithms, and the problem of generating hard instances with solutions.

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Recent experiments demonstrated that local search algorithms (e.g. GSAT) are able to find satisfying assignments for many "hard" Boolean formulas. However, no non-trivial worst-case upper bounds were proved, although many such bounds of the form 2 ffn (ff ! 1 is a constant) are known for other SAT algorithms, e.g. resolution-like algorithms. In the present paper we prove such a bound for a local search algorithm, namely for CSAT. The class of formulas we consider covers most of DIMACS benchmarks, the satisfiability problem for this class of formulas is NP-complete. 1 Introduction Recently there has been an increased interest to local search algorithms for the Boolean satisfiability problem. Though this problem is NP-complete (see e.g. [GaJo]), B. Selman, H. Levesque and D. Mitchell have shown in [SeLeMi] that an algorithm that uses local search can easily handle some of "hard" instances of SAT. They proposed a randomized greedy local search procedure GSAT (see Figure 1) for the Boo...

Abstract. We consider the satisfiability phase transition in skewed random k-SAT distributions. It is known that the random k-SAT model, in which the instance is a set of m k-clauses selected uniformly from the set of all k-clauses over n variables, has a satisfiability phase transition at a certain clause density. The essential feature of the random k-SAT is that positive and negative literals occur with equal probability in a random formula. How does the phase transition behavior change as the relative probability of positive and negative literals changes? In this paper we focus on a distribution in which positive and negative literals occur with different probability. We present empirical evidence for the satisfiability phase transition for this distribution. We also prove an upper bound on the satisfiability threshold and a linear lower bound on the number of literals in satisfying partial assignments of skewed random k-SAT formulas. 1

It has been observed that SAT formulae derived from ATPG problems are efficiently solvable in practise. This seems counter-intuitive since SAT is known to be NP-Complete. This work seeks to explain this paradox. We identify a certain property of circuits which facilitates efficient solution of ATPG-SAT instances arising from them. In addition, we provide both theoretical proofs and empirical evidence to argue that a large fraction of practical VLSI circuits could be expected to have the said property.

Abstract. The probability Psuccess(α, N) that stochastic greedy algorithms successfully solve the random SATisfiability problem is studied as a function of the ratio α of constraints per variable and the number N of variables. These algorithms assign variables according to the unit-propagation (UP) rule in presence

In 1992 B. Selman, H. Levesque and D. Mitchell proposed GSAT, a greedy local search algorithm for the Boolean satisfiability problem. Good performance of this algorithm and its modifications has been demonstrated by many experimental results. In 1993 I. P. Gent and T. Walsh proposed CSAT, a version of GSAT that almost does not use greediness. It has been recently proved that CSAT can find a satisfying assignment for a restricted class of formulas in the time c n , where c ! 2 is a constant. In this paper we prove a lower bound of the order 2 n for GSAT and CSAT. Namely, we construct formulas F n of n variables, such that GSAT or CSAT finds a satisfying assignment for F n only if this assignment or one of its n neighbours is chosen as the initial assignment for the search. 1 Introduction In the past six years there has been an increased interest to local search algorithms for the Boolean satisfiability problem. Though this problem is NP-complete (see e.g. [4]), B. Selman, H. Leve...