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Computational experience with an interior point algorithm on the satisfiability problem

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Abstract

We apply the zero-one integer programming algorithm described in Karmarkar [12] and Karmarkar, Resende and Ramakrishnan [13] to solve randomly generated instances of the satisfiability problem (SAT). The interior point algorithm is briefly reviewed and shown to be easily adapted to solve large instances of SAT. Hundreds of instances of SAT (having from 100 to 1000 variables and 100 to 32,000 clauses) are randomly generated and solved. For comparison, we attempt to solve the problems via linear programming relaxation with MINOS.

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References

  1. C.E. Blair, R.G. Jeroslow and J.K. Lowe, Some results and experiments in programming techniques for propositional logic, Comp. Oper. Res. 5 (1986) 633–645.

    Google Scholar 

  2. T.M. Cavalier, P.M. Pardalos and A.L. Soyster, Modeling and integer programming techniques applied to propositional calculus, Comp. Oper. Res., to appear.

  3. S.A. Cook, The complexity of theorem-proving procedures, in:Proc. 3rd ACM Symp. on the Theory of Computing (1971) pp. 151–158.

  4. M. Davis and H. Putnam. A computing procedure for quantification theory, J. ACM 7 (1960) 201–215.

    Google Scholar 

  5. P.L. Hammer and S. Rudeanu,Boolean Methods in Operations Research and Related Areas (Springer, 1968).

  6. P. Hansen, B. Jaumard and M. Minoux, Algorithm for deriving all logical conclusions implied by a set of Boolean inequalities, Math. Programming 34 (1968) 223–231.

    Google Scholar 

  7. J.N. Hooker, A quantitative approach to logical inference, Decision Support Systems 4 (1988) 45–69.

    Google Scholar 

  8. J.N. Hooker, Generalized resolution and cutting planes, Ann. Oper. Res. 12 (1988) 217–239.

    Google Scholar 

  9. J.N. Hooker, Resolution vs. cutting plane solution of inference problems: Some computational experience, Oper. Res. Lett. 7 (1) (1988) 1–7.

    Google Scholar 

  10. J.N. Hooker and C. Fedjki, Branch-and-cut solution of inference problems in propositional logic, Technical Report 77-88-89, GSIA, Carnegie Mellon University, Pittsburgh, PA 15213 (August 1989), Ann. Math. AI 1 (1990) 123–139.

    Google Scholar 

  11. R.G. Jeroslow, Computation-oriented reductions of predicate to propositional logic, Decision Support Systems 4 (1988) 183–197.

    Google Scholar 

  12. N. Karmarkar, An interior-point approach to NP-complete problems—extended abstract, in:Mathematical Developments Arising from Linear Programming Algorithms, Summer Research Conf. sponsored jointly by AMS, IMS and SIAM, Bowdoin College, Brunswick, Maine (June 1988).

    Google Scholar 

  13. N. Karmarkar, M.G.C. Resende and K.G. Ramakrishnan, An interior-point algorithm for zero-one integer programming, in:13th Int. Symp. on Mathematical Programming, Mathematical Programming Society, Tokyo (September 1988).

    Google Scholar 

  14. N. Karmarkar, M.G.C. Resende and K.G. Ramakrishnan, An interior-point algorithm to solve computationally difficult set covering problems, Technical report, Mathematical Sciences Research Center, AT&T Bell Laboratories, Murray Hill, NJ 07974 (1989).

    Google Scholar 

  15. N. Karmarkar, M.G.C. Resende and K.G. Ramakrishnan, An interior-point approach to the maximum independent set problem in dense random graphs, Technical report, Mathematical Sciences Research Center, AT&T Bell Laboratories, Murray Hill, NJ 07974 (1989).

    Google Scholar 

  16. K. Levenberg, A method for the solution of certain problems in least squares, Quart. Appl. Math. 2 (1944) 164–168.

    Google Scholar 

  17. D. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. 11 (1963) 431–441.

    Google Scholar 

  18. B.A. Murtagh and M.A. Saunders, MINOS 5.0 user's guide, Technical Report SOL 83-20, Dept. of Operations Research, Stanford University, Stanford, CA (1983).

    Google Scholar 

  19. N.J. Nilson,Principles of Artificial Intelligence (Tioga, 1980).

  20. S. Sahni, Computationally related problems, SIAM J. Computing 3 (1974) 262–279.

    Google Scholar 

  21. H.P. Williams, Logical problems and integer programming. Bull. Inst. Math. Appl. 13 (1977) 1820.

    Google Scholar 

  22. H.P. Williams, Linear and integer programming applied to the propositional calculus, Systems Res. Inform. Sci. 2 (1987) 81–100.

    Google Scholar 

  23. R.R. Yager, A mathematical programming approach to inference with the capability of implementing default rules. Int. J. Man-Machine Studies 29 (1988) 685–714.

    Google Scholar 

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Authors and Affiliations

  1. Mathematical Sciences Research Center, AT & T Bell Laboratories, 07974, Murray Hill, NJ, USA

    A. P. Kamath, N. K. Karmarkar, K. G. Ramakrishnan & M. G. C. Resende

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  1. A. P. Kamath
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  2. N. K. Karmarkar
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  3. K. G. Ramakrishnan
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  4. M. G. C. Resende
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Kamath, A.P., Karmarkar, N.K., Ramakrishnan, K.G. et al. Computational experience with an interior point algorithm on the satisfiability problem. Ann Oper Res 25, 43–58 (1990). https://doi.org/10.1007/BF02283686

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