Abstract
In this paper, we propose efficient algorithms to extract minimal unsatisfiable subsets of clauses or variables in unsatisfiable propositional formulas. Such subsets yield unsatisfiable propositional subformulas that become satisfiable when any of their clauses or variables is removed. These subformulas have numerous applications, including proving unsatisfiability and post-infeasibility analysis. The algorithms we propose are based on heuristics, and thus, can be applied to large instances. Furthermore, we show that, in some cases, the minimality of the subformulas can be proven with these algorithms. We also present an original algorithm to find minimum cardinality unsatisfiable subformulas in smaller instances. Finally, we report computational experiments on unsatisfiable instances from various sources, that demonstrate the effectiveness of our algorithms.
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Amaldi E, Pfetsch ME, Trotter LEJ (1999) Some structural and algorithmic properties of the maximum feasible subsystem problem. In: Proceedings of the 7th international IPCO conference on integer programming and combinatorial optimization, June 1999, pp 45–59
Bailey J, Stuckey PJ (2005) Discovery of minimal unsatisfiable subsets of constraints using hitting set dualization. In: Proceedings of the 7th international symposium on practical aspects of declarative languages (PADL05). Lecture notes in computer science, vol 3350. Springer, Berlin, pp 174–186
Battiti R, Protasi M (1998) Approximate algorithms and heuristics for MAX-SAT. In: Ding-zhu D (ed) Handbook of combinatorial optimization, vol 1. Kluwer Academic, Boston
Borchers B, Furman J (1999) A two-phase exact algorithm for MAX-SAT and weighted MAX-SAT problems. J Comb Optim 2:299–306
Bruni R (2003) Approximating minimal unsatisfiable subformulae by means of adaptive core search. Discrete Appl Math 130(2):85–100
Chinneck JW (1997) Finding a useful subset of constraints for analysis in an infeasible linear program. INFORMS J Comput 9(2):164–174
Culberson J (2004) http://web.cs.ualberta.ca/~joe/Coloring/index.html
Dantsin E, Goerdt A, Hirsch EA, Kannan R, Kleinberg J, Papadimitriou C, Raghavan P, Schöning U (2002) A deterministic (2−2/(k+1))n algorithm for k-SAT based on local search. Theor Comput Sci 289(1):69–83
Davis M, Logemann G, Loveland D (1962) A machine program for theorem-proving. Commun ACM 5(7):394–397
Desrosiers C, Galinier P, Hertz A (2008) Efficient algorithms for finding critical subgraphs. Discrete Appl Math 156(2):244–266
Dimacs ftp site (1993) ftp://dimacs.rutgers.edu/pub/challenge/sat/benchmarks/cnf
Fleischner H, Kullmann O, Szeider S (2002) Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theor Comput Sci 289(1):503–516
Franco J, Gu J, Purdom PW, Wah BW (1997) Satisfiability problem: theory and applications. In DIMACS series in discrete mathematics and theoretical computer science, pp 19–152
Galinier P, Hertz A (2007) Solution techniques for the large set covering problem. Discrete Appl Math 155:312–326
Gleeson J, Ryan J (1990) Identifying minimally infeasible subsystems of inequalities. ORSA J Comput 2(1):61–63
Glover F, Laguna M (1997) Tabu search. Kluwer Academic, Boston
Goldberg E, Novikov Y (2000) Berkmin: a fast and robust SAT-solver. In: Design, automation, and test in Europe ’02, March 2000, pp 142–149
Hansen P, Jaumard B (1990) Algorithms for the maximum satisfiability problem. Computing 44(4):279–303
Herrmann F, Hertz A (2002) Finding the chromatic number by means of critical graphs. ACM J Exp Algorithmics 7(10):1–9
Huang J (2005) Mup: a minimal unsatisfiability prover. In: Proceedings of the tenth Asia and South Pacific design automation conference (ASP-DAC-05), pp. 432–437
Kleine Büning H, Zhao X (2002) Polynomial time algorithms for computing a representation for minimal unsatisfiable formulas with fixed deficiency. Inf Process Lett 84(3):147–151
Levesque H, Mitchell D, Selman B (1992) GSAT—a new method for solving hard satisfiability problems. In: Proceedings of the 10th national conference on artificial Intelligence (AAAI-92), pp 440–446
Liffiton MH, Sakallah KA (2005) On finding all minimally unsatisfiable subformulas. In: Proceedings of the 8th international conference on theory and applications of satisfiability testing (SAT-2005). Lecture notes in computer science, vol 3569. Springer, Berlin, pp 173–186
Madigan CF, Malik S, Moskewicz MW, Zhang L, Zhao Y (2001) Chaff: engineering an efficient SAT solver. In: Proceedings of the 38th conference on design automation, June 2001, pp 530–535
Marques Silva JP, Sakallah KA (1999) GRASP: a search algorithm for propositional satisfiability. IEEE Trans Comput 48(5):506–521
Mazure B, Saïs L, Grégoire E (1997) Tabu search for SAT. In: Proceedings of the 14th national conference on artificial intelligence (AAAI-97), pp 281–285
Mazure B, Saïs L, Grégoire E (1998) Boosting complete techniques thanks to local search methods. Ann Math Artif Intell 22(3–4):319–331
Mills P, Tsang E (2000) Guided local search for solving SAT and weighted MAX-SAT problems. J Autom Reas 24(1):205–223
Mneimneh M, Lynce I, Andraus Z, Marques-Silva J, Sakallah K (2005) A branch-and-bound algorithm for extracting smallest minimal unsatisfiable formulas. In: Proceedings of international conference on theory and applications of satisfiability testing, vol 3569, pp 467–474
Oh Y, Mneimneh MN, Andraus ZS, Sakallah KA, Markov IL (2004) AMUSE: a minimally-unsatisfiable subformula extractor. In: Proceedings of the 41st annual conference on design automation. ACM, New York, pp 518–523
Sat benchmarks from automotive product configuration (2003) http://www-sr.informatik.uni-tuebingen.de/~sinz/DC/
Shang Y, Wah BW (1997) Discrete Lagrangian-based search for solving MAX-SAT problems. In: Proceedings of the 15th international joint conference on artificial intelligence, pp 378–383
Zhang H (1997) SATO: an efficient propositional prover. In: Proceedings of international conference on automated deduction (CADE-97), pp 272–275
Zhang L, Malik S (2003) Extracting small unsatisfiable cores from unsatisfiable boolean formulas. In: Sixth international conference on theory and applications of satisfiability testing (SAT2003), May 2003, pp 518–523
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Desrosiers, C., Galinier, P., Hertz, A. et al. Using heuristics to find minimal unsatisfiable subformulas in satisfiability problems. J Comb Optim 18, 124–150 (2009). https://doi.org/10.1007/s10878-008-9142-4
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DOI: https://doi.org/10.1007/s10878-008-9142-4