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Using heuristics to find minimal unsatisfiable subformulas in satisfiability problems

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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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References

  • 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

    Google Scholar 

  • 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

    Google Scholar 

  • Borchers B, Furman J (1999) A two-phase exact algorithm for MAX-SAT and weighted MAX-SAT problems. J Comb Optim 2:299–306

    Article  MATH  MathSciNet  Google Scholar 

  • Bruni R (2003) Approximating minimal unsatisfiable subformulae by means of adaptive core search. Discrete Appl Math 130(2):85–100

    Article  MATH  MathSciNet  Google Scholar 

  • Chinneck JW (1997) Finding a useful subset of constraints for analysis in an infeasible linear program. INFORMS J Comput 9(2):164–174

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Davis M, Logemann G, Loveland D (1962) A machine program for theorem-proving. Commun ACM 5(7):394–397

    Article  MATH  MathSciNet  Google Scholar 

  • Desrosiers C, Galinier P, Hertz A (2008) Efficient algorithms for finding critical subgraphs. Discrete Appl Math 156(2):244–266

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Gleeson J, Ryan J (1990) Identifying minimally infeasible subsystems of inequalities. ORSA J Comput 2(1):61–63

    MATH  Google Scholar 

  • Glover F, Laguna M (1997) Tabu search. Kluwer Academic, Boston

    MATH  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Herrmann F, Hertz A (2002) Finding the chromatic number by means of critical graphs. ACM J Exp Algorithmics 7(10):1–9

    MathSciNet  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Mills P, Tsang E (2000) Guided local search for solving SAT and weighted MAX-SAT problems. J Autom Reas 24(1):205–223

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Chapter  Google Scholar 

  • 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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Authors and Affiliations

  1. Ecole Polytechnique and GERAD, Montreal, Canada

    Christian Desrosiers, Alain Hertz & Sandrine Paroz

  2. Ecole Polytechnique and CRT, Montreal, Canada

    Philippe Galinier

Authors
  1. Christian Desrosiers
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  2. Philippe Galinier
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  3. Alain Hertz
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  4. Sandrine Paroz
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Correspondence to Alain Hertz.

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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

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