zbMATH — the first resource for mathematics

Efficient branch-and-bound algorithms for weighted MAX-2-SAT. (English) Zbl 1216.90073
Summary: MAX-2-SAT is one of the representative combinatorial problems and is known to be NP-hard. Given a set of \(m\) clauses on \(n\) propositional variables, where each clause contains at most two literals and is weighted by a positive real, MAX-2-SAT asks to find a truth assignment that maximizes the total weight of satisfied clauses. In this paper, we propose branch-and-bound exact algorithms for MAX-2-SAT utilizing three kinds of lower bounds. All lower bounds are based on a directed graph that represents conflicts among clauses, and two of them use a set covering representation of MAX-2-SAT. Computational comparisons on benchmark instances disclose that these algorithms are highly effective in reducing the number of search tree nodes as well as the computation time.

90C27 Combinatorial optimization
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
Full Text: DOI
[1] Alsinet, T., Manya, F., Planes, J.: Improved branch and bound algorithms for Max-SAT. In: Sixth International Conference on Theory and Applications of Satisfiability Testing, pp. 408–415 (2003)
[2] Alsinet T., Manyà F., Planes J.: A Max-SAT solver with lazy data structures. In: Lemaître, C., Reyes, C.A., Gonzalez, J.A. (eds) Advances in Artificial Intelligence–IBERAMIA 2004. Lecture Notes in Artificial Intelligence, vol. 3315, pp. 334–342. Springer, Heidelberg (2004)
[3] Amini M.M., Alidaee B., Kochenberger G.A.: A scatter search approach to unconstrained quadratic binary programs. In: Corne, D., Dorigo, M., Glover, F. (eds) New Ideas in Optimization, pp. 317–329. McGraw-Hill, London (1999)
[4] Aspvall B., Plass M.R., Tarjan R.E.: A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Inf. Process. Lett. 8, 121–123 (1979) · Zbl 0398.68042 · doi:10.1016/0020-0190(79)90002-4
[5] Bansal, N., Raman, V.: Upper bounds for MaxSat: Further improved. In: ISAAC ’99: Proceedings of the 10th International Symposium on Algorithms and Computation, pp. 247–258. Springer-Verlag, London (1999) · Zbl 0971.68069
[6] Bonami P., Minoux M.: Exact MAX-2SAT solution via lift-and-project closure. Oper. Res. Lett. 34, 387–393 (2006) · Zbl 1133.90373 · doi:10.1016/j.orl.2005.07.001
[7] Borchers B., Furman J.: A two-phase exact algorithm for MAX-SAT and weighted MAX-SAT problems. J. Combin. Optim. 2, 299–306 (1999) · Zbl 0954.90026 · doi:10.1023/A:1009725216438
[8] Boros, E.: Private communication via e-mail exchange (2007)
[9] Boros, E., Hammer, P.L.: A max-flow approach to improved roof-duality in quadratic 0–1 minimization. RUTCOR Research Report RRR 15-1989, RUTCOR (1989)
[10] Boros E., Hammer P.L.: Pseudo-Boolean optimization. Discrete Appl. Math. 123, 155–225 (2002) · Zbl 1076.90032 · doi:10.1016/S0166-218X(01)00341-9
[11] Boros E., Crama Y., Hammer P.L.: Upper-bounds for quadratic 0–1 maximization. Oper. Res. Lett. 9, 73–79 (1990) · Zbl 0699.90073 · doi:10.1016/0167-6377(90)90044-6
[12] Boros E., Crama Y., Hammer P.L.: Chvátal cuts and odd cycle inequalities in quadratic 0–1 optimization. SIAM J. Discrete Math. 5, 163–177 (1992) · Zbl 0761.90069 · doi:10.1137/0405014
[13] Boros, E., Hammer, P.L., Tavares, G.: Preprocessing of unconstrained quadratic binary optimization. RUTCOR Research Report RRR 10-2006, RUTCOR (2006)
[14] Boros E., Hammer P.L., Sun R., Tavares G.: A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO). Discrete Optim. 5, 501–529 (2008) · Zbl 1170.90454 · doi:10.1016/j.disopt.2007.02.001
[15] Bourjolly, J.-M., Hammer, P.L., Pulleyblank, W.R., Simeone, B.: Combinatorial methods for bounding quadratic pseudo-Boolean functions. RUTCOR Research Report RRR 27-1989, RUTCOR (1989)
[16] Bourjolly J.-M., Hammer P.L., Pulleyblank W.R., Simeone B.: Boolean-combinatorial bounding of maximum 2-satisfiability. In: Balci, O., Sharda, R., Zenios, S.A. (eds) Computer Science and Operations Research: New Developments in their Interfaces, pp. 23–42. Pergamon Press, Oxford (1992)
[17] Cherkassky B.V., Goldberg A.V.: On implementing the push-relabel method for the maximum flow problem. Algorithmica 19, 390–410 (1997) · Zbl 0898.68029 · doi:10.1007/PL00009180
[18] Cormen T.H., Leiserson C.E., Rivest R.L., Stein C.: Introduction to Algorithms. MIT Press, Cambridge (2001) · Zbl 1047.68161
[19] Davis M., Putnam H.: A computing procedure for quantification theory. J. ACM 7, 201–215 (1960) · Zbl 0212.34203 · doi:10.1145/321033.321034
[20] de Givry S., Larrosa J., Meseguer P., Schiex T.: Solving Max-SAT as weighted CSP. In: Rossi, F. (eds) Principles and Practice of Constraint Programming (CP 2003). Lecture Notes in Computer Science, vol. 2833, pp. 363–376. Springer, Heidelberg (2003) · Zbl 1273.68368
[21] de Klerk E., Warners J.P.: Semidefinite programming approaches for MAX-2-SAT and MAX-3-SAT: Computational perspectives. In: Pardalos, P.M., Migdalas, A., Burkard, R.E. (eds) Combinatorial and Global Optimization. Series on Applied Optimization, vol. 14, pp. 161–176. World Scientific Publishers, Singapore (2002) · Zbl 1029.90053
[22] Eén N., Sörensson N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds) Theory and Applications of Satisfiability Testing–SAT 2003. Lecture Notes in Computer Science, vol. 2919, pp. 502–518. Springer, Heidelberg (2004) · Zbl 1204.68191
[23] Eén N., Sörensson N.: Translating pseudo-Boolean constraints into SAT. J. Satisfiability Boolean Model. Comput. 2, 1–25 (2006) · Zbl 1116.68083
[24] Even S., Itai A., Shamir A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5, 691–703 (1976) · Zbl 0358.90021 · doi:10.1137/0205048
[25] Garey M.R., Johnson D.S., Stockmeyer L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1, 237–267 (1976) · Zbl 0338.05120 · doi:10.1016/0304-3975(76)90059-1
[26] Gilmore P.C., Gomory R.E.: A linear programming approach to the cutting stock problem. Oper. Res. 9, 849–859 (1961) · Zbl 0096.35501 · doi:10.1287/opre.9.6.849
[27] Glover F., Kochenberger G., Alidaee B.: Adaptive memory tabu search for binary quadratic programs. Manage. Sci. 44, 336–345 (1998) · Zbl 0989.90072 · doi:10.1287/mnsc.44.3.336
[28] Glover F., Kochenberger G., Alidaee B., Amini M.: Tabu search with critical event memory: An enhanced application for binary quadratic programs. In: Voss, S., Martello, S., Osman, I.H., Roucairol, C. (eds) Meta-Heuristics: Advances and Trends in Local Search Paradigms for Optimization, pp. 93–109. Kluwer Academic Publishers, Boston (1999) · Zbl 0972.90053
[29] Glover F., Alidaee B., Rego C., Kochenberger G.: One-pass heuristics for large-scale unconstrained binary quadratic problems. Eur. J. Oper. Res. 137, 272–287 (2002) · Zbl 1030.90074 · doi:10.1016/S0377-2217(01)00209-0
[30] Goldberg A.V., Tarjan R.E.: A new approach to the maximum-flow problem. J. ACM 35, 921–940 (1988) · Zbl 0661.90031 · doi:10.1145/48014.61051
[31] Goldberg, E., Novikov, Y.: BerkMin: A fast and robust SAT solver. In: Design Automation and Test in Europe (DATE 2002), pp. 142–149 (2002) · Zbl 1121.68106
[32] Gramm, J., Niedermeier, R.: Faster exact solutions for MAX2SAT. In: Conference on Algorithms and Complexity, pp. 174–186 (2000) · Zbl 0971.68598
[33] Hammer P.L., Hansen P., Simeone B.: Roof duality, complementation and persistency in quadratic 0–1 optimization. Math. Program. 28, 121–155 (1984) · Zbl 0574.90066 · doi:10.1007/BF02612354
[34] Hansen P., Jaumard B.: Algorithms for the maximum satisfiability problem. Computing 44, 279–303 (1990) · Zbl 0716.68077 · doi:10.1007/BF02241270
[35] Heras F., Larrosa J., Oliveras A.: MiniMaxSAT: a new weighted Max-SAT solver. In: Marques-Silva, J., Sakallah, K.A. (eds) Theory and Applications of Satisfiability Testing–SAT 2007. Lecture Notes in Computer Science, vol. 4501, pp. 41–55. Springer, Heidelberg (2007)
[36] Hirsch, E.A.: A new algorithm for MAX-2-SAT. In: STACS 2000: 17th Annual Symposium on Theoretical Aspects of Computer Science, pp. 65–73 (2000) · Zbl 0959.68047
[37] Johnson D.S.: Local optimization and the traveling salesman problem. In: Paterson, M.S. (eds) Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 443, pp. 446–461. Springer, Heidelberg (1990) · Zbl 0766.90079
[38] Joy, S., Mitchell, J.E., Borchers, B.: Solving MAX-SAT and weighted MAX-SAT problems using branch-and-cut. Technical report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 (1998) · Zbl 0889.68074
[39] Kirkpatrick S., Selman B.: Critical behavior in the satisfiability of random Boolean expressions. Science 264, 1297–1301 (1994) · Zbl 1226.68097 · doi:10.1126/science.264.5163.1297
[40] Kochenberger G., Glover F., Alidaee B., Lewis K.: Using the unconstrained quadratic program to model and solve Max 2-SAT problems. Int. J. Oper. Res. 1, 89–100 (2005) · Zbl 1100.90059 · doi:10.1504/IJOR.2005.007435
[41] Koga, Y.: Efficient branch-and-bound algorithms for weighted MAX-2-SAT. Master’s thesis, Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University (2006)
[42] Li C.-M., Manya F., Planes J.: New inference rules for Max-SAT. J. Artif. Intell. Res. 30, 321–359 (2007) · Zbl 1182.68254
[43] Lourenço H.R., Martin O.C., Stützle T.: Iterated local search. In: Glover, F., Kochenberger, G.A. (eds) Handbook of Metaheuristics, pp. 321–353. Kluwer Academic Publishers, Boston (2003) · Zbl 1116.90412
[44] Luby M., Ragde P.: A bidirectional shortest-path algorithm with good average-case behavior. Algorithmica 4, 551–567 (1989) · Zbl 0681.68068 · doi:10.1007/BF01553908
[45] Mahajan Y.S., Fu Z., Malik S.: Zchaff2004: An efficient SAT solver. In: Hoos, H.H., Mitchell, D.G. (eds) Theory and Applications of Satisfiability Testing–SAT 2004. Lecture Notes in Computer Science, vol. 3542, pp. 360–375. Springer, Heidelberg (2005) · Zbl 1122.68610
[46] Martin O., Otto S.W., Felten E.W.: Large-step Markov chains for the traveling salesman problem. Complex Syst. 5, 299–326 (1991) · Zbl 0736.90063
[47] Merz P., Freisleben B.: Greedy and local search heuristics for unconstrained binary quadratic programming. J. Heuristics 8, 197–213 (2002) · Zbl 1013.90100 · doi:10.1023/A:1017912624016
[48] Miyashiro R., Matsui T.: A polynomial-time algorithm to find an equitable home-away assignment. Oper. Res. Lett. 33, 235–241 (2005) · Zbl 1177.90175 · doi:10.1016/j.orl.2004.06.004
[49] Miyashiro R., Matsui T.: Semidefinite programming based approaches to the break minimization problem. Comput. Oper. Res. 33, 1975–1982 (2006) · Zbl 1090.90154 · doi:10.1016/j.cor.2004.09.030
[50] Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proceedings of the 39th Design Automation Conference (DAC 2001), pp. 530–535 (2001)
[51] Niedermeier R., Rossmanith P.: New upper bounds for maximum satisfiability. J. Algorithms 36, 63–88 (2000) · Zbl 0959.68049 · doi:10.1006/jagm.2000.1075
[52] Resende M.G.C., Pitsoulis L.S., Pardalos P.M.: Approximate solution of weighted MAX-SAT problems using GRASP. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 35, 393–405 (1997) · Zbl 0889.68139
[53] Selman, B., Levesque, H.J., Mitchell, D.: A new method for solving hard satisfiability problems. In: Rosenbloom, P., Szolovits, P. (eds.) Proceedings of the Tenth National Conference on Artificial Intelligence, pp. 440–446. AAAI Press, Menlo Park (1992)
[54] Selman, B., Kautz, H.A., Cohen, B.: Noise strategies for improving local search. In: Proceedings of the Twelfth National Conference on Artificial Intelligence (AAAI’94), pp. 337–343. Seattle (1994)
[55] Shen, H., Zhang, H.: An empirical study of MAX-2-SAT phase transitions. In: LICS’03 Workshop on Typical Case Complexity and Phase Transitions, June (2003) · Zbl 1179.68148
[56] Shen, H., Zhang, H.: Improving exact algorithms for MAX-2-SAT. In: Eighth International Symposium on Artificial Intelligence and Mathematics, January (2004) · Zbl 1086.68058
[57] Shen, H., Zhang, H.: Study of lower bound functions for MAX-2-SAT. In: 19th National Conference on Artificial Intelligence (AAAI), pp. 185–190 (2004)
[58] Simeone, B.: A generalized consensus approach to nonlinear 0–1 minimization. Technical Report CORR 38/79, Department of Combinatorics and Optimization, University of Waterloo (1979) · Zbl 0428.90045
[59] Simeone, B.: Quadratic 0–1 Programming, Boolean Functions, and Graphs. Ph.D. dissertation, Department of Combinatorics and Optimization, University of Waterloo (1979)
[60] Smyth K., Hoos H.H., Stützle T.: Iterated robust tabu search for MAX-SAT. In: Xiang, Y., Brahim, C.-D. Advances in Artificial Intelligence: 16th Conference of the Canadian Society for Computational Studies of Intelligence (AI 2003). Lecture Notes in Artificial Intelligence, vol. 2671, pp. 129–144. Springer, Heidelberg (2003)
[61] Tavares, G.: Max2SatGen: A generator of weighted MAX-2-SAT formulas. RUTCOR, Rutgers University (2005)
[62] Tompkins D.A.D., Hoos H.H.: UBCSAT: An implementation and experimentation environment for SLS algorithms for SAT and MAX-SAT. In: Hoos, H.H., Mitchell, D.G. (eds) Theory and Applications of Satisfiability Testing–SAT 2004. Lecture Notes in Computer Science, vol. 3542, pp. 306–320. Springer, Heidelberg (2005) · Zbl 1122.68620
[63] Trick, M.A.: A schedule-then-break approach to sports timetabling. In: PATAT ’00: Selected papers from the Third International Conference on Practice and Theory of Automated Timetabling III, pp. 242–253. Springer, Heidelberg (2001) · Zbl 0982.68531
[64] Wallace, R.J.: Enhancing maximum satisfiability algorithms with pure literal strategies. In: McCalla, G. (ed.) Advances in Artificial Intelligence: 11th Biennial Conference of the Canadian Society for Computational Studies of Intelligence (AI96). Lecture Notes in Computer Science, vol. 1081 (1996)
[65] Wallace, R.J., Freuder, E.C.: Comparative studies of constraint satisfaction and Davis-Putnam algorithms for maximum satisfiability problems. In: Johnson, D.S., Trick, M.A. (eds.) Cliques, Coloring, and Satisfiability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26, pp. 587–615 (1996) · Zbl 0859.68072
[66] Xing Z., Zhang W.: MaxSolver: An efficient exact algorithm for (weighted) maximum satisfiability. Artif. Intell. 164, 47–80 (2005) · Zbl 1132.68716 · doi:10.1016/j.artint.2005.01.004
[67] Yagiura M., Ibaraki T.: Analyses on the 2 and 3-flip neighborhoods for the MAX SAT. J. Combin. Optim. 3, 95–114 (1999) · Zbl 0955.90119 · doi:10.1023/A:1009873324187
[68] Yagiura M., Ibaraki T.: Efficient 2 and 3-flip neighborhood search algorithms for the MAX SAT: Experimental evaluation. J. Heuristics 7, 423–442 (2001) · Zbl 1041.68091 · doi:10.1023/A:1011306011437
[69] Yannakakis M.: On the approximation of maximum satisfiability. J. Algorithms 17, 475–502 (1994) · Zbl 0820.68056 · doi:10.1006/jagm.1994.1045
[70] Zhang, H.: SATO: An efficient propositional prover. In: Proceedings of the International Conference on Automated Deduction (CADE’97), pp. 272–275 (1997)
[71] Zhang H., Shen H., Manyà F.: Exact algorithms for MAX-SAT. Electronic Notes in Theoretical Computer Science 86(1), 190–203 (2003) · Zbl 1261.68073 · doi:10.1016/S1571-0661(04)80663-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.