Abstract
A matched formula is a CNF formula whose incidence graph admits a matching which matches a distinct variable to every clause. We study phase transition in a context of matched formulas and their generalization of biclique satisfiable formulas. We have performed experiments to find a phase transition of property “being matched” with respect to the ratio \(m/n\) where \(m\) is the number of clauses and \(n\) is the number of variables of the input formula \(\varphi \). We compare the results of experiments to a theoretical lower bound which was shown by Franco and Van Gelder [11]. Any matched formula is satisfiable, and it remains satisfiable even if we change polarities of any literal occurrences. Szeider [17] generalized matched formulas into two classes having the same property—var-satisfiable and biclique satisfiable formulas. A formula is biclique satisfiable if its incidence graph admits covering by pairwise disjoint bounded bicliques. Recognizing if a formula is biclique satisfiable is NP-complete. In this paper we describe a heuristic algorithm for recognizing whether a formula is biclique satisfiable and we evaluate it by experiments on random formulas. We also describe an encoding of the problem of checking whether a formula is biclique satisfiable into SAT and we use it to evaluate the performance of our heuristic.
This research was supported by Charles University project UNCE/SCI/004 and SVV project number 260 453. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the programme “Projects of Large Research, Development, and Innovations Infrastructures” (CESNET LM2015042), is greatly appreciated.
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References
Metacentrum grid computing, June 2017. https://metavo.metacentrum.cz/en/
Aharoni, R., Linial, N.: Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. J. Comb. Theory Ser. A 43(2), 196–204 (1986). https://doi.org/10.1016/0097-3165(86)90060-9. http://www.sciencedirect.com/science/article/pii/0097316586900609
Audemard, G., Simon, L.: The glucose sat solver, June 2017. http://www.labri.fr/perso/lsimon/glucose/
Bollobás, B.: Modern Graph Theory. GTM, vol. 184. Springer, New York (1998). https://doi.org/10.1007/978-1-4612-0619-4
Cheeseman, P., Kanefsky, B., Taylor, W.M.: Where the really hard problems are. In: Proceedings of the 12th International Joint Conference on Artificial Intelligence, IJCAI 1991, vol. 1, pp. 331–337. Morgan Kaufmann Publishers Inc., San Francisco (1991). http://dl.acm.org/citation.cfm?id=1631171.1631221
Chromý, M., Kučera, P.: Phase transition in matched formulas and a heuristic for biclique satisfiability. ArXiv e-prints, August 2018. https://arxiv.org/abs/1808.01774
Connamacher, H.S., Molloy, M.: The satisfiability threshold for a seemingly intractable random constraint satisfaction problem. CoRR abs/1202.0042 (2012). http://arxiv.org/abs/1202.0042
Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing, STOC 1971, pp. 151–158. ACM, New York (1971). https://doi.org/10.1145/800157.805047
Dubois, O., Boufkhad, Y., Mandler, J.: Typical random 3-SAT formulae and the satisfiability threshold. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2000, pp. 126–127. Society for Industrial and Applied Mathematics, Philadelphia (2000). http://dl.acm.org/citation.cfm?id=338219.338243
Eén, N., Sörensson, N.: The minisat, June 2017. http://minisat.se/
Franco, J., Van Gelder, A.: A perspective on certain polynomial-time solvable classes of satisfiability. Discrete Appl. Math. 125(2–3), 177–214 (2003). https://doi.org/10.1016/S0166-218X(01)00358-4. http://www.sciencedirect.com/science/article/pii/S0166218X01003584
Gent, I.P., Walsh, T.: The SAT phase transition. In: Proceedings of the 11th European Conference on Artificial Intelligence, pp. 105–109. Wiley (1994)
Heydari, M.H., Morales, L., Shields Jr., C.O., Sudborough, I.H.: Computing cross associations for attack graphs and other applications. In: 2007 40th Annual Hawaii International Conference on System Sciences, HICSS 2007, p. 270b, January 2007. https://doi.org/10.1109/HICSS.2007.141
Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)
Lovász, L., Plummer, M.D.: Matching Theory. North-Holland, Amsterdam (1986)
Mitchell, D., Selman, B., Levesque, H.: Hard and easy distributions of SAT problems. In: Proceedings of the 10th National Conference on Artificial Intelligence (AAAI 1992), San Jose, CA, USA, pp. 459–465 (1992)
Szeider, S.: Generalizations of matched CNF formulas. Ann. Math. Artif. Intell. 43(1), 223–238 (2005). https://doi.org/10.1007/s10472-005-0432-6
Tovey, C.A.: A simplified NP-complete satisfiability problem. Discrete Appl. Math. 8(1), 85–89 (1984). https://doi.org/10.1016/0166-218X(84)90081-7. http://www.sciencedirect.com/science/article/pii/0166218X84900817
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Chromý, M., Kučera, P. (2019). Phase Transition in Matched Formulas and a Heuristic for Biclique Satisfiability. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_10
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