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On optimizing the satisfiability (SAT) problem

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Abstract

The satisfiability (SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient optimization algorithms for finding a solution for a satisfiableCNF formula. A new formulation, theUniversal SAT problem model, which transforms the SAT problem on Boolean space into an optimization problem on real space has been developed. Many optimization techniques, such as the steepest descent method, Newton's method, and the coordinate descent method, can be used to solve theUniversal SAT problem. In this paper, we prove that, when the initial solution is sufficiently close to the optimal solution, the steepest descent method has a linear convergence ratio β<1, Newton's method has a convergence ratio of order tow, and the convergence ratio of the coordinate descent method is approximately (1-β/m) for theUniversal SAT problem withm variables. An algorithm based on the coordinate descent method for theUniversal SAT problem is also presented in this paper.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Calgary, T2N 1N4, Calgary, Canada

    Gu Jun

  2. Department of Computer Science, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Gu Jun

  3. Department of Computer Software, The University of Aizu, Aizu-Wakamatsu, 965-80, Fukushima, Japan

    Gu Qianping

  4. Department of Computer Science, University of Minnesota, 55455, Minneapolis, MN, U.S.A.

    Du Dingzhu

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  1. Gu Jun
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  2. Gu Qianping
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Correspondence to Gu Jun.

Additional information

This work was supported in part by NSERC Strategic Grant MEF0045793 and NSERC Research Grant OGP0046423.

A part of an early version of this paper was published in IEEE Transactions on Computers, Vol. 45, No. 2, 1996.

For the biography ofGU Jun, please refer to p. 90 of this issue.

GU Qianping received the B.S. degree from Shandong University, China, M.S. degree from Ibaraki University, Japan, and Ph.D. degree from Tohoku University, Japan, all in computer science, in 1982, 1985, and 1988, respectively. He is currently an Associate Professor in the Department of Computer Software, the University of Aizu, Japan. He was with the Institute of Software, Chinese Academy of Sciences, Beijing, China, and the Department of Electrical and Computer Engineering, the University of Calgary, Canada. His research interests include algorithms, computational complexity, machine learning, parallel processing, and optimization. He is a member of ACM, IEEE Computer Society, and IEICE of Japan.

DU Dingzhu received his M.S. degree in 1982 from the Institute of Applied Mathematics, Chinese Academy of Sciences, and his Ph.D. degree in Computer Science in 1985 from the University of California at Santa Barbara. He visited MSRI in 1985, MIT in 1986, and Princeton University in 1990. Currently he is a tenure faculty at the Department of Computer Science, University of Minnesota, and a research Professor at the Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing. He has published eight books and over 100 journal papers. His research interests are computational complexity, analysis and design of algorithms, combinatorial optimization, computational geometry, communication networks, and linear and nonlinear programming. In 1986, with Xiangsun Zhang, he established the convergence of Rosen's gradient projection method and in 1990, with Frank Huang, he proved the Gilbert-Pollak conjecture on Steiner ratio.

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Gu, J., Gu, Q. & Du, D. On optimizing the satisfiability (SAT) problem. J. Comput. Sci. & Technol. 14, 1–17 (1999). https://doi.org/10.1007/BF02952482

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