Handbook of Combinatorial Optimization pp 77-148 | Cite as

# Approximate Algorithms and Heuristics for *MAX-SAT*

Chapter

## Abstract

In the Maximum Satisfiability (*MAX-SAT*) problem one is given a Boolean formula in conjunctive normal form, i.e., as a conjunction of clauses, each clause being a disjunction. The task is to find an assignment of truth values to the variables that satisfies the maximum number of clauses.

## Keywords

Local Search Integer Linear Programming Performance Ratio Approximate Algorithm Truth Assignment
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]D. Achlioptas, L. M. Kirousis, E. Kranakis, and
**D**. Krinzac,*Rigorous results for random (2+p)-SAT*, Proc. Work. on Randomized Algorithms in Sequential, Parallel and Distributed Computing, Santorini, Greece, 1997.Google Scholar - [2]P. Alimonti,
*New local search approximation techniques for maximum generalized satisfiability problems*, Proc. Second Italian Conf. on Algorithms and Complexity, Rome, 1994, pp. 40–53.Google Scholar - [3]S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy,
*Proof verification and hardness of approximation problems*, Proc. 33rd Annual IEEE Symp. on Foundations of Computer Science, IEEE Computer Society, 1992, pp. 14–23.Google Scholar - [4]
*S. Arora and S. Safra*Probabilistic checking of proofs:*a new*characterization of NP*Proc. 33rd Annual IEEE Symp. on Foundations of Computer Science, IEEE Computer Society, 1992, pp. 2–13.*Google Scholar - [5]
*T. Asano*Approximation algorithms for MAX-SAT: Yannakakis vs. G oemans- Williamson*Proc. 3rd Israel Symp. on the Theory of Computing and Systems, Ramat Gan, Israel, 1997, pp. 24–37.*Google Scholar - [6]
*T. Asano, T. Ono, and T. Hirata*Approximation algorithms for the maximum satisfiability problem*Proc. 5th Scandinavian Work. on Algorithms Theory, 1996, pp. 110–111.*Google Scholar - [7]P. Asirelli, M. de Santis, and A. M.rtelli,
*Integrity constraints in logic databases*, Journal of Logic Programming**3**(1985), 221–232.Google Scholar - [8]G. Ausiello, P. Crescenzi, and M. Protasi, Approximate solution of NP optimization problems, Theoretical Computer Science 150 (1995), 1-55.CrossRefzbMATHMathSciNetGoogle Scholar
- [9]G. Ausiello, A. D’Atri, and M. Protasi,
*Lattice theoretic properties of NP-complete problems*, Fundamenta Informaticae**4**(1981), 83–94.zbMATHMathSciNetGoogle Scholar - [10]
*G. Ausiello and M. Protasi*Local search reducibility and approximability of NP-optimization problems*Information Processing Letters***54***(1995), 73–79.*Google Scholar - [11]R.Battiti,
*Reactive search: Toward self-tuning heuristics*, Modern Heuristic Search Methods (V.J. Rayward-Smith,I.**H**. Osman, C.R. Reeves, and G.D. Smith, eds.),John Wiley and Sons, 1996, pp. 61–83.Google Scholar - [12]
*R. Battiti and M. Protasi*Reactive search a history-sensitive heuristic for MAX-SAT*ACM Journal of Experimental Algorithmics***2***(1997), no. 2**http://www.jea.acm.org/.*Google Scholar - [13]
*Solving MAX-SAT with non-oblivious functions and history-based heuristics*,Satisfiability Problem: Theory and Applications, DI-MACS: Series in Discrete Mathematics and Theoretical Computer Science, no. 35, AMS and ACM Press, 1997.Google Scholar - [14]R. Battiti and G. Tecchiolli,
*The reactive tabu search*, ORSA Journal on Computing**6**(1994), no. 2, 126–140.zbMATHGoogle Scholar - [15]C.E. Blair, R.G. Jeroslow, and J.K. Lowe,
*Some results and experiments in programming for propositional logic*, Computers and Operations Research**13**(1986), no. 5, 633–645.CrossRefzbMATHMathSciNetGoogle Scholar - [16]
*M. Boehm and E. Speckenmeyer***A**fast parallel sat solver - efficient workload balancing*Annals of Mathematics and Artificial Intelligence***17***(1996), 381–400.*Google Scholar - [17]
*A. Broder, A. Frieze, and E. Upfal*On the satisfiability and maximum satisfiability of random 3-CNF formulas*Proc. of the 4th Annual ACM-SIAM Symp. on Discrete Algorithms, 1993.*Google Scholar - [18]M. Buro and H. Kleine Buening,
*Report on a SAT competition*, EATCS Bulletin**49**(1993), 143–151.zbMATHGoogle Scholar - [19]S. Chakradar, V. Agrawal, and M. Bushnell,
*Neural net and boolean satisfiability model of logic circuits*, IEEE Design and Test of Computers (1990), 54–57.Google Scholar - [20]
*M.-T. Chao and J. Franco*Probabilistic analysis of two heuristics for the 3-satisfiability problem*SIAM Journal on Computing***15***(1986), 1106–1118.*Google Scholar - [21]
*J. Chen, D. Friesen, and H. Zheng*Tight bound on Johnson’s algorithm for MAX-SAT*Proc. 12th Annual IEEE Conf. on Computational Complexity, Ulm, Germany, 1997, pp. 274–281.*Google Scholar - [22]
*J. Cheriyan, W*..**H***Cunningham, T. Tuncel, and Y. Wang*A linear programming and rounding approach to MAX 2-SAT*Proc. of the Second DIMACS Algorithm Implementation Challenge on Cliques, Coloring and Satisfiability (M. Trick and D. S. Johson, eds.), DIMACS Series on Discrete Mathematics and Theoretical Computer Science, no. 26, 1996, pp. 395–414.*Google Scholar - [23]V. Chvatal and B. Reed,
*Mick gets some (the odds are on his side)*, Proc. 33th Ann. IEEE Symp. on Foundations of Computer Science, IEEE Computer Society, 1992, pp. 620–627.Google Scholar - [24]V. Chvâtal and E. Szemerédi,
*Many hard examples for resolution*, Journal of the ACM**35**(1988), 759–768.CrossRefzbMATHGoogle Scholar - [25]S.A. Cook,
*The complexity of theorem-proving procedures*,Proc. of the Third Annual ACM Symp. on the Theory of Computing, 1971, pp. 151-158.Google Scholar - [26]
*S.A. Cook and D.G. Mitchell*Finding hard instances of the satisfiability problem: a survey*Satisfiability Problem: Theory and Applications*D.-Z. Du, J. Gu, and P.M. Pardalos, eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol.*35*,AMS and ACM Press,*1997.*Google Scholar - [27]
*P*. Crescenzi and A. Panconesi,*Completeness in approximation classes*,Information and Computation**93***(1991)*,*241–262.*Google Scholar - [28]
*M. Davis, G. Logemann, and D. Loveland*A machine program for theorem proving*Communications of the ACM 5*(1962) 394–397.CrossRefzbMATHMathSciNetGoogle Scholar - [29]
*M. Davis and H. Putnam*A computing procedure for quantification theory*Journal of the ACM 7*(1960) 201–215.CrossRefMathSciNetGoogle Scholar - [30]D. Du, J. Gu, and P.M. Pardalos (Eds.),
*Satisfiability problem: Theory and applications*,DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol.*35*,AMS and ACM Press,*1997.*Google Scholar - [31]
*O*. Dubois and Y. Boufkhad,*A general upper bound for the satisfiability threshold of random r-SAT formulas*,Tech. report, LAFORIA, CNRSUniv. Paris*6*,*1996.*Google Scholar - [32]
*U*. Feige and M.X. Goemans,*Approximating the value of two proper proof systems*,*with applications to MAX-2SAT and MAX-DICUT*,Proc. of the Third Israel Symp. on Theory of Computing and Systems,*1995*,*pp. 182–189.*Google Scholar - [33]J. Franco and M. Paull,
*Probabilistic analysis of the davis-putnam procedure for solving the satisfiability problem*,Discrete Applied Mathematics 5*(1983)*,*77–87.*Google Scholar - [34]
*A. Frieze and S. Suen*Analysis of two simple heuristics on a random instance of k-SAT*Journal of Algorithms 20*(1996) 312–355.CrossRefzbMATHMathSciNetGoogle Scholar - [35]
*H. Gallaire, J. Minker, and J. M. Nicolas*Logic and databases: a deductive approach*Computing Surveys 16*(1984)*no. 2*153–185.CrossRefMathSciNetGoogle Scholar - [36]
*I.P. Gent and T. Walsh*An empirical analysis of search in gsat*Journal of Artificial Intelligence Research 1*(1993) 47–59.Google Scholar - [37],
*Towards an understanding of hill-climbing procedures for SAT*,Proc. of the Eleventh National Conf. on Artificial Intelligence, AAAI Press/The MIT Press,*1993*,*pp. 28–33.*Google Scholar - [38]
*F. Glover*Tabu search - part I*ORSA Journal on Computing 1*(1989)*no. 3*190–260.MathSciNetGoogle Scholar - [39]
*M.X. Goemans and***D.P.***Williamson*New 3/4-approximation algorithms for the maximum satisfiability problem*SIAM Journal on Discrete Mathematics***7***(1994), no. 4, 656–666*.Google Scholar - [40]Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
*Journal of the ACM***42***(1995), no. 6, 1115–1145*Google Scholar - [41]
*A. Goerdt*A threshold for unsatisfiability*Journal of Computer and System Sciences***53***(1996), 469–486*CrossRefMathSciNetGoogle Scholar - [42]
*J. Gu*Efficient local search for*very*large-scale satisfiability problem*ACM SIGART Bulletin 3 (1992), no. 1, 8–12*Google Scholar - [43]
*Global optimization for satisfiability (SAT) problem*,IEEE Transactions on Data and Knowledge Engineering**6**(1994), no. 3, 361381.Google Scholar - [44]J. Gu, Q.-P. Gu, and D.-Z.Du,
*Convergence properties of optimization algorithms for the SAT problem*, IEEE Transactions on Computers**45**(1996), no. 2, 209–219.CrossRefzbMATHGoogle Scholar - [45]J. Gu, P.W. Purdom, J. Franco, and B.W. Wah, Algorithms for the satisfiability (SAT) problem: A survey, Satisfiability Problem: Theory and Applications (D.-Z. Du, J. Gu, and P.M. Pardalos, eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 35, AMS and ACM Press, 1997.Google Scholar
- [46]
*J. Gu and R. Puri*Asynchronous circuit synthesis with boolean satisfiability*IEEE Transactions on Computer-Aided Design of Integrated Circuits***14***(1995), no. 8, 961–973*Google Scholar - [47]
*P.L. Hammer, P. Hansen, and B. Simeone*Roof duality complementation and persistency in quadratic 0–1 optimization*Mathematical Programming***28***(1984), 121–155*Google Scholar - [48]
*P. Hansen and*.**B***Jaumard*Algorithms for the maximum satisfiability problem*Computing***44***(1990), 279–303.*Google Scholar - [49]
*J.N. Hooker*Resolution vs. cutting plane solution of inference problems: some computational experience*Operations Research Letters***7***(1988), no. 1, 1–7.*CrossRefMathSciNetGoogle Scholar - [50]J. Hâstad,
*Some optimal inapproximability results*, Proc. 28th Annual ACM Symp. on Theory of Computing, El Paso, Texas, 1997, pp. 1–10.Google Scholar - [51]B. Jaumard, M. Stan, and J. Desrosiers,
*Tabu search and a quadratic relaxation for the satisfiability problem*,Proc. of the Second DIMACS Algorithm Implementation Challenge on Cliques, Coloring and Satisfiability (M. Trick and D. S. Johson, eds.), DIMACS Series on Discrete Mathematics and Theoretical Computer Science, no. 26, 1996, pp. 457477.Google Scholar - [52]D.S. Johnson,
*Approximation algorithms for combinatorial problems*, Journal of Computer and System Sciences**9**(1974), 256–278.CrossRefMathSciNetGoogle Scholar - [53]D.S. Johnson and M. Trick (Eds.),
*Cliques*,*coloring*,*and satisfiability: Second DIMA CS implementation challenge*, vol. 26, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, no. 26, AMS, 1996.Google Scholar - [54]J.L. Johnson,
*A neural network approach to the 3-satisfiability problem*, Journal of Parallel and Distributed Computing**6**(1989), 435–449.CrossRefGoogle Scholar - [55]A. Kamath, R. Motwani, K. Palem, and P. Spirakis,
*Tail bounds for occupancy and the satisfiability threshold conjecture*, Random Structures and Algorithms**7**(1995), 59–80.CrossRefzbMATHMathSciNetGoogle Scholar - [56]A.P. Kamath, N.K. Karmarkar, K.G. Ramakrishnan, and M.G. Re-sende,
*Computational exprience with an interior point algorithm on the satisfiability problem*, Annals of Operations Research**25**(1990), 43–58.CrossRefzbMATHMathSciNetGoogle Scholar - [57]
*A continuous approach to inductive inference*,Mathematical programming**57**(1992), 215–238.Google Scholar - [58]
**H**. Karloff and U. Zwick,*A 7/8-approximation algorithm for MAX 3SAT?*,Proc. of the 38th Annual IEEE Symp. on Foundations of Computer Science, IEEE Computer Society, 1997, in press.Google Scholar - [59]S. Khanna, R.Motwani, M.Sudan, and U.Vazirani,
*On syntactic versus computational views of approximability*, Proc. 35th Ann. IEEE Symp. on Foundations of Computer Science, IEEE Computer Society, 1994, pp. 819–836.Google Scholar - [60]S. Kirkpatrick, C.D. Gelatt Jr., and M.P. Vecchi,
*Optimization by simulated annealing*, Science**220**(1983), 671–680.CrossRefzbMATHMathSciNetGoogle Scholar - [61]S. Kirkpatrick and B. Selman,
*Critical behavior in the satisfiability of random boolean expressions*, Science**264**(1994), 1297–1301.CrossRefzbMATHMathSciNetGoogle Scholar - [62]
*L.M. Kirousis, E. Kranakis, and D. Krizanc*Approximating the unsatisfiability threshold of random formulas*Proc. of the Fourth Annual European Symp. on Algorithms (Barcelona), Springer-Verlag, September 1996, pp. 27–38.*Google Scholar - [63]E. Koutsoupias and C.H. Papadimitriou,
*On the greedy algorithm for satisfiability*, Information Processing Letters**43**(1992), 53–55.CrossRefzbMATHMathSciNetGoogle Scholar - [64]
*O. Kullmann and H. Luckhardt*Deciding propositional tautologies: Algorithms and their complexity*Tech. Report 1596, JohannWolfgang Goethe-Univ., Fachbereich Mathematik, Frankfurt, Germany, January 1997.*Google Scholar - [65]D.W. Loveland,
*Automated theorem proving: A logical basis*, North-Holland, 1978.Google Scholar - [66]
*S. Minton, M. D. Johnston, A. B. Philips, and P. Laird*Solving large-scale constraint satisfaction and scheduling problems using a heuristic repair method*Proc. of the 8th National Conf. on Artificial Intelligence (AAAI-90), 1990, pp. 17–24.*Google Scholar - [67]
*D. Mitchell, B. Selman, and H. Levesque*Hard and easy distributions of SAT problems*Proc. of the 10th National Conf. on Artificial Intelligence (AAAI-92) (San Jose, Ca), July 1992, pp. 459–465.*Google Scholar - [68]R. Motwani and P. Raghavan,
*Randomized algorithms*, Cambridge University Press, New York, 1995.zbMATHGoogle Scholar - [69]T.A. Nguyen, W.A. Perkins, T.J. Laffrey, and D. Pecora,
*Checking an expert system knowledge base for consistency and completeness*, Proc. of the International Joint Conf. on Artificial Intelligence (Los Altos, CA ), 1985, pp. 375–378.Google Scholar - [70]P. Nobili and A. Sassano,
*Strengthening lagrangian bounds for the MAX-SAT problem*,Tech. Report 96–230, Institut fuer Informatik, Koln Univ., Germany, 1996, Proc. of the Work. on the Satisfiability Problem, Siena, Italy (J. Franco and G. Gallo and**H**. Kleine Buening, Eds.).Google Scholar - [71]P. Orponen and
**H**. Mannila, On*approximation preserving reductions: complete problems and robust measures*, Tech. Report C-1987–28, Dept. of Computer Science, Univ. of Helsinki, 1987.Google Scholar - [72]
*C. H. Papadimitriou*On selecting a satisfying truth assignment (extended abstract)*Proc. of the 32th Annual Symp. on Foundations of Computer Science, 1991, pp. 163–169.*Google Scholar - [73]C.H. Papadimitriou and K. Steiglitz,
*Combinatorial optimization*,*algorithms and complexity*, Prentice-Hall, NJ, 1982.Google Scholar - [74]R. Puri and J. Gu,
*A BDD SAT solver for satisfiability testing: an industrial case study*, Annals of Mathematics and Artificial Intelligence 17 (1996), no. 3–4, 315–337.CrossRefzbMATHMathSciNetGoogle Scholar - [75]M.G.C. Resende and T. A. Feo,
*A grasp for satisfiability*, Proc. of the Second DIMACS Algorithm Implementation Challenge on Cliques, Coloring and Satisfiability (M. Trick and D. S. Johson, eds.), DIMACS Series on Discrete Mathematics and Theoretical Computer Science, no. 26, 1996, pp. 499–520.Google Scholar - [76]
*M.G.C. Resende, L.S. Pitsoulis, and P.M. Pardalos*Approximate solution of weighted MAX-SAT problems using GRASP*Satisfiability Problem: Theory and Applications, DIMACS: Series in Discrete Mathematics and Theoretical Computer Sc ience, no. 35, 1997.*Google Scholar - [77]J. A. Robinson,
*A machine-oriented logic based on the resolution principle*, Journal of the ACM 12 (1965), 23–41.CrossRefzbMATHGoogle Scholar - [78]
*B. Selman and H. Kautz*Domain-independent extensions to GSAT: Solving large structured satisfiability problems*Proc. of the International Joint Conf. on Artificial Intelligence, 1993, pp. 290–295.*Google Scholar - [79]
*B. Selman and H.A. Kautz*An empirical study of greedy local search for satisfiability testing*Proc. of the 11th National Conf. on Artificial Intelligence (AAAI-93) (Washington, D. C.), 1993.*Google Scholar - [80]B. Selman, H.A. Kautz, and B. Cohen,
*Local search strategies for satisfiability testing*, Proc. of the Second DIMACS Algorithm Implementation Challenge on Cliques, Coloring and Satisfiability (M. Trick and D. S. Johson, eds.), DIMACS Series on Discrete Mathematics and Theoretical Computer Science, no. 26, 1996, pp. 521–531.Google Scholar - [81]
*B. Selman, H. Levesque, and D. Mitchell*A new method for solving hard satisfiability problems*Proc. of the 10th National Conf. on Artificial Intelligence (AAAI-92) (San Jose, Ca), July 1992, pp. 440–446.*Google Scholar - [82]W.M. Spears,
*Simulated annealing for hard satisfiability problems*, Proc. of the Second DIMACS Algorithm Implementation Challenge on Cliques, Coloring and Satisfiability (M. Trick and D. S. Johnson, eds.), DIMACS Series on Discrete Mathematics and Theoretical Computer Science, no. 26, 1996, pp. 533–555.Google Scholar - [83]L. Trevisan,
*Approximating satisfiable satisfiability problems*, Proc. of the 5th Annual European Symp. on Algorithms, Graz, Springer Verlag, 1997, pp. 472–485.Google Scholar - [84]
*M. Yannakakis*On the approximation of maximum satisfiability*Journal of Algorithms***17**(1994), 475–502.CrossRefzbMATHMathSciNetGoogle Scholar

## Copyright information

© Kluwer Academic Publishers 1998