Abstract
We present novel approaches to detect cardinality constraints expressed in CNF. The first approach is based on a syntactic analysis of specific data structures used in SAT solvers to represent binary and ternary clauses, whereas the second approach is based on a semantic analysis by unit propagation. The syntactic approach computes an approximation of the cardinality constraints AtMost-1 and AtMost-2 constraints very fast, whereas the semantic approach has the property to be generic, i.e. it can detect cardinality constraints AtMost-k for any k, at a higher computation cost. Our experimental results suggest that both approaches are efficient at recovering AtMost-1 and AtMost-2 cardinality constraints.
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References
Aloul, F.A., Ramani, A., Markov, I.L., Sakallah, K.A.: Generic ilp versus specialized 0-1 ilp: An update. In: Pileggi, L.T., Kuehlmann, A. (eds.) ICCAD, pp. 450–457. ACM (2002)
Ansótegui, C., Larrubia, J., Li, C.M., Manyà, F.: Exploiting multivalued knowledge in variable selection heuristics for sat solvers. Ann. Math. Artif. Intell. 49(1-4), 191–205 (2007)
Ansótegui, C., Manyà, F.: Mapping problems with finite-domain variables to problems with boolean variables. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 1–15. Springer, Heidelberg (2005)
Ansótegui Gil, C.J.: Complete SAT solvers for Many-Valued CNF Formulas. Ph.D. thesis. University of Lleida (2004)
Asín, R., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E.: Cardinality networks and their applications. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 167–180. Springer, Heidelberg (2009)
Bailleux, O., Boufkhad, Y.: Efficient cnf encoding of boolean cardinality constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 108–122. Springer, Heidelberg (2003)
Barahona, P., Hölldobler, S., Nguyen, V.-H.: Representative Encodings to Translate Finite CSPs into SAT. In: Simonis, H. (ed.) CPAIOR 2014. LNCS, vol. 8451, pp. 251–267. Springer, Heidelberg (2014)
Barth, P.: Linear 0-1 inequalities and extended clauses. In: Voronkov, A. (ed.) LPAR 1993. LNCS, vol. 698, pp. 40–51. Springer, Heidelberg (1993)
Ben-Haim, Y., Ivrii, A., Margalit, O., Matsliah, A.: Perfect hashing and cnf encodings of cardinality constraints. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 397–409. Springer, Heidelberg (2012)
Biere, A.: Lingeling, plingeling and treengeling entering the sat competition 2013. In: Balint, A., Belov, A., Heule, M., Järvisalo, M. (eds.) Proceedings of SAT Competition 2013. Solver and Benchmark Descriptions, vol. B-2013-1, pp. 51–52. University of Helsinki, Department of Computer Science Series of Publications B (2013)
Chen, J.C.: A new sat encoding of the at-most-one constraint. In: Proc. of the Tenth Int. Workshop of Constraint Modelling and Reformulation (2010)
Cook, W., Coullard, C., Turán, G.: On the complexity of cutting-plane proofs. Discrete Applied Mathematics 18(1), 25–38 (1987)
Eén, N., Sörensson, N.: Translating pseudo-boolean constraints into sat. JSAT 2(1-4), 1–26 (2006)
Frisch, A., Giannaros, P.: Sat encodings of the at-most-k constraint: Some old, some new, some fast, some slow. In: Proceedings of the 9th International Workshop on Constraint Modelling and Reformulation, ModRef 2010 (2010)
Fu, Z., Malik, S.: Extracting logic circuit structure from conjunctive normal form descriptions. In: VLSI Design, pp. 37–42. IEEE Computer Society (2007)
Van Gelder, A., Spence, I.: Zero-one designs produce small hard sat instances. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 388–397. Springer, Heidelberg (2010)
Gent, I.P., Nightingale, P.: A new encoding of alldifferent into sat. In: Proc. 3rd International Workshop on Modelling and Reformulating Constraint Satisfaction Problems, pp. 95–110 (2004)
Gent, I., Prosser, P., Smith, B.: A 0/1 encoding of the gaclex constraint for pairs of vectors. In: ECAI 2002 workshop W9: Modelling and Solving Problems with Constraints. University of Glasgow (2002)
Haken, A.: The intractability of resolution. Theoretical Computer Science 39(0), 297–308 (1985)
Hölldobler, S., Nguyen, V.H.: On SAT-Encodings of the At-Most-One Constraint. In: Katsirelos, G., Quimper, C.G. (eds.) Proc. The Twelfth International Workshop on Constraint Modelling and Reformulation, Uppsala, Sweden, September 16-20, pp. 1–17 (2013)
Hooker, J.N.: Generalized resolution and cutting planes. Ann. Oper. Res. 12(1-4), 217–239 (1988)
Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J., Bohlinger, J. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Springer, US (1972)
Klieber, W., Kwon, G.: Efficient cnf encoding for selecting 1 from n objects. In: International Workshop on Constraints in Formal Verification (2007)
van Lambalgen, M.: 3MCard 3MCard A Lookahead Cardinality Solver. Master’s thesis. Delft University of Technology (2006)
Le Berre, D.: Exploiting the real power of unit propagation lookahead. Electronic Notes in Discrete Mathematics 9, 59–80 (2001)
Manthey, N., Heule, M.J.H., Biere, A.: Automated reencoding of boolean formulas. In: Biere, A., Nahir, A., Vos, T. (eds.) HVC. LNCS, vol. 7857, pp. 102–117. Springer, Heidelberg (2013)
Manthey, N., Steinke, P.: Quadratic direct encoding vs. linear order encoding, a one-out-of-n transformation on cnf. In: Proceedings of the First International Workshop on the Cross-Fertilization Between CSP and SAT, CSPSAT11 (2011)
Martins, R., Manquinho, V.M., Lynce, I.: Parallel search for maximum satisfiability. AI Commun. 25(2), 75–95 (2012)
Mikša, M., Nordström, J.: Long proofs of (Seemingly) simple formulas. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 122–138. Springer, Heidelberg (2014)
Ostrowski, R., Grégoire, É., Mazure, B., Sais, L.: Recovering and exploiting structural knowledge from cnf formulas. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 185–199. Springer, Heidelberg (2002)
Sinz, C.: Towards an optimal cnf encoding of boolean cardinality constraints. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 827–831. Springer, Heidelberg (2005)
Spence, I.: sgen1: A generator of small but difficult satisfiability benchmarks. ACM Journal of Experimental Algorithmics 15 (2010)
Walsh, T.: Sat v csp. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 441–456. Springer, Heidelberg (2000)
Weaver, S.: Satisfiability Advancements Enabled by State Machines. Ph.D. thesis. University of Cincinnati (2012)
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Biere, A., Le Berre, D., Lonca, E., Manthey, N. (2014). Detecting Cardinality Constraints in CNF. In: Sinz, C., Egly, U. (eds) Theory and Applications of Satisfiability Testing – SAT 2014. SAT 2014. Lecture Notes in Computer Science, vol 8561. Springer, Cham. https://doi.org/10.1007/978-3-319-09284-3_22
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