Efficient branch-and-bound algorithms for weighted MAX-2-SAT

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.

[1]  Etienne de Klerk,et al.  Semidefinite Programming Approaches for MAX-2-SAT and MAX-3-SAT: computational perspectives , 2002 .

[2]  Peter L. Hammer,et al.  Boolean-Combinatorial Bounding of Maximum 2-Satisfiability , 1992, Computer Science and Operations Research.

[3]  Toshihide Ibaraki,et al.  Analyses on the 2 and 3-Flip Neighborhoods for the MAX SAT , 1999, J. Comb. Optim..

[4]  Michel Minoux,et al.  Exact MAX-2SAT solution via lift-and-project closure , 2006, Oper. Res. Lett..

[5]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[6]  Y. Crama,et al.  Upper-bounds for quadratic 0-1 maximization , 1990 .

[7]  Hantao Zhang,et al.  Improving exact algorithms for MAX-2-SAT , 2005, Annals of Mathematics and Artificial Intelligence.

[8]  Panos M. Pardalos,et al.  Approximate solution of weighted MAX-SAT problems using GRASP , 1996, Satisfiability Problem: Theory and Applications.

[9]  Miguel Toro,et al.  Advances in Artificial Intelligence — IBERAMIA 2002 , 2002, Lecture Notes in Computer Science.

[10]  Bart Selman,et al.  Noise Strategies for Improving Local Search , 1994, AAAI.

[11]  Endre Boros,et al.  Chvátal Cuts and ODD Cycle Inequalities in Quadratic 0 - 1 Optimization , 1992, SIAM J. Discret. Math..

[12]  Thomas Stützle,et al.  Iterated Robust Tabu Search for MAX-SAT , 2003, Canadian Conference on AI.

[13]  Hiroshi Nagamochi,et al.  EFFICIENT BRANCH-AND-BOUND ALGORITHMS FOR WEIGHTED MAX-2-SAT , 2007 .

[14]  Teresa Alsinet,et al.  A Max-SAT Solver with Lazy Data Structures , 2004, IBERAMIA.

[15]  Andrew V. Goldberg,et al.  On Implementing the Push—Relabel Method for the Maximum Flow Problem , 1997, Algorithmica.

[16]  Sharad Malik,et al.  Zchaff2004: An Efficient SAT Solver , 2004, SAT (Selected Papers.

[17]  Weixiong Zhang,et al.  MaxSolver: An efficient exact algorithm for (weighted) maximum satisfiability , 2005, Artif. Intell..

[18]  Niklas Sörensson,et al.  Translating Pseudo-Boolean Constraints into SAT , 2006, J. Satisf. Boolean Model. Comput..

[19]  Toshihide Ibaraki,et al.  Efficient 2 and 3-Flip Neighborhood Search Algorithms for the MAX SAT: Experimental Evaluation , 2001, J. Heuristics.

[20]  Bernd Freisleben,et al.  Greedy and Local Search Heuristics for Unconstrained Binary Quadratic Programming , 2002, J. Heuristics.

[21]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[22]  F. Glover,et al.  Adaptive Memory Tabu Search for Binary Quadratic Programs , 1998 .

[23]  Fred W. Glover,et al.  One-pass heuristics for large-scale unconstrained binary quadratic problems , 2002, Eur. J. Oper. Res..

[24]  Edward W. Felten,et al.  Large-Step Markov Chains for the Traveling Salesman Problem , 1991, Complex Syst..

[25]  Simon de Givry,et al.  Solving Max-SAT as Weighted CSP , 2003, CP.

[26]  Prabhakar Ragde,et al.  A bidirectional shortest-path algorithm with good average-case behavior , 1989, Algorithmica.

[27]  Pierre Hansen,et al.  Algorithms for the maximum satisfiability problem , 1987, Computing.

[28]  John E. Mitchell,et al.  Solving MAX-SAT and weighted MAX-SAT problems using branch-and-cut , 1998 .

[29]  Hantao Zhang,et al.  An Empirical Study of MAX-2-SAT Phase Transitions , 2003, Electron. Notes Discret. Math..

[30]  Alon Itai,et al.  On the Complexity of Timetable and Multicommodity Flow Problems , 1976, SIAM J. Comput..

[31]  Eugene Goldberg,et al.  BerkMin: A Fast and Robust Sat-Solver , 2002 .

[32]  Bahram Alidaee,et al.  A scatter search approach to unconstrained quadratic binary programs , 1999 .

[33]  Felip Manyà,et al.  Exact Algorithms for MAX-SAT , 2003, FTP.

[34]  Venkatesh Raman,et al.  Upper Bounds for MaxSat: Further Improved , 1999, ISAAC.

[35]  Richard J. Wallace,et al.  Enhancing Maximum Satisfiablility Algorithms with Pure Literal Strategies , 1996, Canadian Conference on AI.

[36]  R. Gomory,et al.  A Linear Programming Approach to the Cutting-Stock Problem , 1961 .

[37]  Hilary Putnam,et al.  A Computing Procedure for Quantification Theory , 1960, JACM.

[38]  Mihalis Yannakakis,et al.  On the approximation of maximum satisfiability , 1992, SODA '92.

[39]  David S. Johnson,et al.  Local Optimization and the Traveling Salesman Problem , 1990, ICALP.

[40]  F. Glover,et al.  Using the unconstrained quadratic program to model and solve Max 2-SAT problems , 2005 .

[41]  Edward A. Hirsch,et al.  A New Algorithm for MAX-2-SAT , 2000, STACS.

[42]  Michael A. Trick A Schedule-Then-Break Approach to Sports Timetabling , 2000, PATAT.

[43]  S Kirkpatrick,et al.  Critical Behavior in the Satisfiability of Random Boolean Expressions , 1994, Science.

[44]  Endre Boros,et al.  Pseudo-Boolean optimization , 2002, Discret. Appl. Math..

[45]  Holger H. Hoos,et al.  UBCSAT: An Implementation and Experimentation Environment for SLS Algorithms for SAT & MAX-SAT , 2004, SAT.

[46]  Helena Ramalhinho Dias Lourenço,et al.  Iterated Local Search , 2001, Handbook of Metaheuristics.

[47]  Hector J. Levesque,et al.  A New Method for Solving Hard Satisfiability Problems , 1992, AAAI.

[48]  Rolf Niedermeier,et al.  Faster Exact Solutions for MAX2SAT , 2000, CIAC.

[49]  Hantao Zhang,et al.  Study of Lower Bound Functions for MAX-2-SAT , 2004, AAAI.

[50]  Rolf Niedermeier,et al.  New Upper Bounds for Maximum Satisfiability , 2000, J. Algorithms.

[51]  Sharad Malik,et al.  Chaff: engineering an efficient SAT solver , 2001, Proceedings of the 38th Design Automation Conference (IEEE Cat. No.01CH37232).

[52]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas , 1979, Inf. Process. Lett..

[53]  Pierre Hansen,et al.  Roof duality, complementation and persistency in quadratic 0–1 optimization , 1984, Math. Program..

[54]  Andrew V. Goldberg,et al.  A new approach to the maximum flow problem , 1986, STOC '86.

[55]  Hantao Zhang,et al.  SATO: An Efficient Propositional Prover , 1997, CADE.

[56]  Teresa Alsinet,et al.  Improved branch and bound algorithms for Max-SAT , 2003 .

[57]  Endre Boros,et al.  Preprocessing of unconstrained quadratic binary optimization , 2006 .

[58]  Eugene C. Freuder,et al.  Anytime algorithms for constraint satisfaction and SAT problems , 1996, SGAR.

[59]  Felip Manyà,et al.  New Inference Rules for Max-SAT , 2007, J. Artif. Intell. Res..

[60]  F. Glover,et al.  Tabu Search with Critical Event Memory: An Enhanced Application for Binary Quadratic Programs , 1999 .

[61]  Tomomi Matsui,et al.  A polynomial-time algorithm to find an equitable home-away assignment , 2005, Oper. Res. Lett..

[62]  Niklas Sörensson,et al.  An Extensible SAT-solver , 2003, SAT.

[63]  Endre Boros,et al.  A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO) , 2008, Discret. Optim..

[64]  Albert Oliveras,et al.  MiniMaxSat: A New Weighted Max-SAT Solver , 2007, SAT.

[65]  Brian Borchers,et al.  A Two-Phase Exact Algorithm for MAX-SAT and Weighted MAX-SAT Problems , 1998, J. Comb. Optim..

[66]  Tomomi Matsui,et al.  Semidefinite programming based approaches to the break minimization problem , 2006, Comput. Oper. Res..

[67]  Armando Tacchella,et al.  Theory and Applications of Satisfiability Testing: 6th International Conference, Sat 2003, Santa Margherita Ligure, Italy, May 5-8 2003: Selected Revised Papers (Lecture Notes in Computer Science, 2919) , 2004 .