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.
Similar content being viewed by others
References
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.
T.M. Cavalier, P.M. Pardalos and A.L. Soyster, Modeling and integer programming techniques applied to propositional calculus, Comp. Oper. Res., to appear.
S.A. Cook, The complexity of theorem-proving procedures, in:Proc. 3rd ACM Symp. on the Theory of Computing (1971) pp. 151–158.
M. Davis and H. Putnam. A computing procedure for quantification theory, J. ACM 7 (1960) 201–215.
P.L. Hammer and S. Rudeanu,Boolean Methods in Operations Research and Related Areas (Springer, 1968).
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.
J.N. Hooker, A quantitative approach to logical inference, Decision Support Systems 4 (1988) 45–69.
J.N. Hooker, Generalized resolution and cutting planes, Ann. Oper. Res. 12 (1988) 217–239.
J.N. Hooker, Resolution vs. cutting plane solution of inference problems: Some computational experience, Oper. Res. Lett. 7 (1) (1988) 1–7.
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.
R.G. Jeroslow, Computation-oriented reductions of predicate to propositional logic, Decision Support Systems 4 (1988) 183–197.
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).
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).
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).
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).
K. Levenberg, A method for the solution of certain problems in least squares, Quart. Appl. Math. 2 (1944) 164–168.
D. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. 11 (1963) 431–441.
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).
N.J. Nilson,Principles of Artificial Intelligence (Tioga, 1980).
S. Sahni, Computationally related problems, SIAM J. Computing 3 (1974) 262–279.
H.P. Williams, Logical problems and integer programming. Bull. Inst. Math. Appl. 13 (1977) 1820.
H.P. Williams, Linear and integer programming applied to the propositional calculus, Systems Res. Inform. Sci. 2 (1987) 81–100.
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1007/BF02283686