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Algorithms for the Satisfiability (SAT) Problem: A Survey
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1996
"... . 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, computeraided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, compute ..."
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Cited by 127 (3 self)
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. 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, computeraided 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...
A Discrete LagrangianBased GlobalSearch Method for Solving Satisfiability Problems
 Journal of Global Optimization
, 1998
"... Satisfiability is a class of NPcomplete problems that model a wide range of realworld applications. These problems are difficult to solve because they have many local minima in their search space, often trapping greedy search methods that utilize some form of descent. In this paper, we propose a n ..."
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Cited by 60 (7 self)
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Satisfiability is a class of NPcomplete problems that model a wide range of realworld applications. These problems are difficult to solve because they have many local minima in their search space, often trapping greedy search methods that utilize some form of descent. In this paper, we propose a new discrete Lagrangemultiplierbased globalsearch method for solving satisfiability problems. We derive new approaches for applying Lagrangian methods in discrete space, show that equilibrium is reached when a feasible assignment to the original problem is found, and present heuristic algorithms to look for equilibrium points. Instead of restarting from a new starting point when a search reaches a local trap, the Lagrange multipliers in our method provide a force to lead the search out of a local minimum and move it in the direction provided by the Lagrange multipliers. One of the major advantages of our method is that it has very few algorithmic parameters to be tuned by users, and the se...
Asynchronous Circuit Synthesis with Boolean Satisfiability
, 1995
"... Asynchronous circuits are widely used in many real time applications such as digital communication and computer systems. The design of complex asynchronous circuits is a difficult and errorprone task. An adequate synthesis method will significantly simplify the design and reduce errors. In this pap ..."
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Cited by 20 (1 self)
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Asynchronous circuits are widely used in many real time applications such as digital communication and computer systems. The design of complex asynchronous circuits is a difficult and errorprone task. An adequate synthesis method will significantly simplify the design and reduce errors. In this paper, we present a general and efficient partitioning approach to the synthesis of asynchronous circuits from general Signal Transition Graph (STG) specifications. The method partitions a large signal transition graph into smaller and manageable subgraphs which significantly reduces the complexity of asynchronous circuit synthesis. Experimental results of our partitioning approach with large number of practical industrial asynchronous circuit benchmarks are presented. They show that, compared to the existing asynchronous circuit synthesis techniques, this partitioning approach achieves many orders of magnitude of performance improvements in terms of computing time, in addition to the reduced c...
TraceBased Methods for Solving Nonlinear Global Optimization and Satisfiability Problems
 J. of Global Optimization
, 1996
"... . In this paper we present a method called NOVEL (Nonlinear Optimization via External Lead) for solving continuous and discrete global optimization problems. NOVEL addresses the balance between global search and local search, using a trace to aid in identifying promising regions before committing to ..."
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Cited by 15 (5 self)
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. In this paper we present a method called NOVEL (Nonlinear Optimization via External Lead) for solving continuous and discrete global optimization problems. NOVEL addresses the balance between global search and local search, using a trace to aid in identifying promising regions before committing to local searches. We discuss NOVEL for solving continuous constrained optimization problems and show how it can be extended to solve constrained satisfaction and discrete satisfiability problems. We first transform the problem using Lagrange multipliers into an unconstrained version. Since a stable solution in a Lagrangian formulation only guarantees a local optimum satisfying the constraints, we propose a global search phase in which an aperiodic and bounded trace function is added to the search to first identify promising regions for local search. The trace generates an informationbearing trajectory from which good starting points are identified for further local searches. Taking only a sm...
Global Search Methods For Solving Nonlinear Optimization Problems
, 1997
"... ... these new methods, we develop a prototype, called Novel (Nonlinear Optimization Via External Lead), that solves nonlinear constrained and unconstrained problems in a unified framework. We show experimental results in applying Novel to solve nonlinear optimization problems, including (a) the lear ..."
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Cited by 15 (1 self)
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... these new methods, we develop a prototype, called Novel (Nonlinear Optimization Via External Lead), that solves nonlinear constrained and unconstrained problems in a unified framework. We show experimental results in applying Novel to solve nonlinear optimization problems, including (a) the learning of feedforward neural networks, (b) the design of quadraturemirrorfilter digital filter banks, (c) the satisfiability problem, (d) the maximum satisfiability problem, and (e) the design of multiplierless quadraturemirrorfilter digital filter banks. Our method achieves better solutions than existing methods, or achieves solutions of the same quality but at a lower cost.
An Improved Semidefinite Programming Relaxation for the Satisfiability Problem
, 2002
"... The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there ..."
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Cited by 14 (4 self)
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The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NPcomplete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variable X is maximized (or minimized) subject to linear constraints on the elements of X and the additional constraint that X be positive semidefinite. We are interested in the application of SDP to satisfiability problems, and in particular in how SDP can be used to detect unsatisfiability. In this paper we introduce a new SDP relaxation for the satisfiability problem. This SDP relaxation arises from the recently introduced paradigm of “higher liftings” for constructing semidefinite programming relaxations of discrete optimization problems.
Lagrangian Techniques for Solving a Class of ZeroOne Integer Linear Programs
 In Proc. Computer Software and Applications Conf
, 1995
"... We consider a class of zeroone integer programming feasibility problems (01 ILPF problems) in which the coefficients of variables can be integers, and the objective is to find an assignment of binary variables so that all constraints are satisfied. We propose a Lagrangian formulation in the contin ..."
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Cited by 13 (4 self)
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We consider a class of zeroone integer programming feasibility problems (01 ILPF problems) in which the coefficients of variables can be integers, and the objective is to find an assignment of binary variables so that all constraints are satisfied. We propose a Lagrangian formulation in the continuous space and develop a gradient search in this space. By using two counteracting forces, one performing gradient search in the primal space (of the original variables) and the other in the dual space (of the Lagrangian variables) , we show that our search algorithm does not get trapped in local minima and reaches equilibrium only when a feasible assignment to the original problem is found. We present experimental results comparing our method with backtracking and local search (based on random restarts). Our results show that 01 ILPF problems of reasonable sizes can be solved by an order of magnitude faster than existing methods. 1 Introduction In this paper, we study the search of a feas...
Combining Cellular Genetic Algorithms and Local Search for Solving Satisfiability Problems
 In Proceedings of Tenth IEEE International Conference on Tools with Artificial Intelligence
, 1998
"... A new parallel hybrid method for solving the satisfiability problem that combines cellular genetic algorithms and the random walk (WSAT ) strategy of GSAT is presented. The method, called CGWSAT , uses a cellular genetic algorithm to perform a global search on a random initial population of candidat ..."
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Cited by 10 (0 self)
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A new parallel hybrid method for solving the satisfiability problem that combines cellular genetic algorithms and the random walk (WSAT ) strategy of GSAT is presented. The method, called CGWSAT , uses a cellular genetic algorithm to perform a global search on a random initial population of candidate solutions and a local selective generation of new strings. Global search is specialized in local search by adopting the WSAT strategy. CGWSAT has been implemented on a Meiko CS2 parallel machine using a twodimensional cellular automaton as parallel computation model. The algorithm has been tested on randomly generated problems and some classes of problems from the DIMACS test set. 1. Introduction The satisfiability problem (SAT ) consists in finding a truth assignment that makes a boolean expression true. Satisfiability plays a central role in a broad range of fields such as artificial intelligence, mathematical logic, computer vision, VLSI design, databases, automated reasoning, comp...
Solving MAXSAT and Weighted MAXSAT Problems Using BranchandCut
, 1998
"... We describe a branch and cut algorithm for both MAXSAT and weighted MAXSAT. This algorithm uses the GSAT procedure as a primal heuristic. At each nodewe solve a linear programming (LP) relaxation of the problem. Two styles of separating cuts are added: resolution cuts and odd cycle inequalities. W ..."
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Cited by 5 (2 self)
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We describe a branch and cut algorithm for both MAXSAT and weighted MAXSAT. This algorithm uses the GSAT procedure as a primal heuristic. At each nodewe solve a linear programming (LP) relaxation of the problem. Two styles of separating cuts are added: resolution cuts and odd cycle inequalities. We compare our algorithm to an extension of the Davis Putnam Loveland (EDPL) algorithm and a SemiDefinite Programming (SDP) approach. Our algorithm is more effective than EDPL on some problems, notably MAX2SAT. EDPL and SDP are more effective on some other classes of problems.
The Theory And Applications Of Discrete Constrained Optimization Using Lagrange Multipliers
, 2000
"... In this thesis, we present a new theory of discrete constrained optimization using Lagrange multipliers and an associated firstorder search procedure (DLM) to solve general constrained optimization problems in discrete, continuous and mixedinteger space. The constrained problems are general in the ..."
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Cited by 4 (0 self)
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In this thesis, we present a new theory of discrete constrained optimization using Lagrange multipliers and an associated firstorder search procedure (DLM) to solve general constrained optimization problems in discrete, continuous and mixedinteger space. The constrained problems are general in the sense that they do not assume the differentiability or convexity of functions. Our proposed theory and methods are targeted at discrete problems and can be extended to continuous and mixedinteger problems by coding continuous variables using a floatingpoint representation (discretization). We have characterized the errors incurred due to such discretization and have proved that there exists upper bounds on the errors. Hence, continuous and mixedinteger constrained problems, as well as discrete ones, can be handled by DLM in a unified way with bounded errors.