Abstract
We propose an algorithm using a spectral method, and analyze its average-case performance for MAX2SAT in the planted solution model. In [16], they proposed a distribution \(\mathcal{G}_{n,p,r}\) for MAX2SAT in the planted solution model, as well as a message-passing algorithm. They showed that it solves, \(\textbf{whp}\), MAX2SAT on \(\mathcal{G}_{n,p,r}\) for rather dense formulas, i.e., the expected number of clauses is \(\Omega(n^{1.5}\sqrt{\log n})\). In this paper, we propose an algorithm using a spectral method and a variant of message-passing algorithms, and show that it solves, \(\textbf{whp}\), MAX2SAT on \(\mathcal{G}_{n,p,r}\) for sparser formulas, i.e., the expected number of clauses is Ω(nlogn).
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References
Alon, N., Kahale, N.: A spectral technique for coloring random 3-colorable graphs. SIAM J. Comput. 26(6), 1733–1748 (1997)
Böttcher, J.: Coloring sparse random k-colorable graphs in polynomial expected time. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 156–167. Springer, Heidelberg (2005)
Coja-Oghlan, A.: A spectral heuristic for bisecting random graphs. In: Proceedings of 16th ACM-SIAM Symp. on Discrete Algorithms (SODA 2005), pp. 850–859 (2005)
Coja-Oghlan, A.: An Adaptive Spectral Heuristic for Partitioning Random Graphs. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 691–702. Springer, Heidelberg (2006)
Coja-Oghlan, A., Taraz, A.: Colouring Random Graphs in Expected Polynomial Time. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 511–522. Springer, Heidelberg (2003)
Coja-Oghlan, A., Goerdt, A., Lanka, A., Schädlich, F.: Techniques from combinatorial approximation algorithms yield efficient algorithms for random 2k-SAT. Theoretical Computer Science 329, 1–45 (2004)
Coja-Oghlan, A., Krivelevich, M., Vilenchik, D.: Why almost all k-colorable graphs are easy. In: Proceedings of 24th International Symp. on Theoretical Aspects of Computer Science (STACS 2007), pp. 121–132 (2007)
Condon, A., Karp, R.M.: Algorithms for graph partitioning on the planted partition model. Random Struct. Algorithms 18(2), 116–140 (2001)
Håstad, J.: Some optimal inapproximability results. J. of the ACM 48, 798–859 (2001)
Feige, U., Mossel, E., Vilenchik, D.: Complete convergence of message passing algorithms for some satisfiability problems. In: Proceedings of APPROX-RANDOM 2006, pp. 339–350 (2006)
Feige, U., Vilenchik, D.: A local search algorithm for 3SAT. Technical report of the Weizmann Institute of science (2004)
Flaxman, A.: A spectral technique for random satisfiable 3CNF formulas. In: Proceedings of 14th ACM-SIAM Symp. on Discrete Algorithms, SODA 20 03, pp. 357–363 (2003), See also http://www.math.cmu.edu./adf/research/spectralSat
Krivelevich, M., Vilenchik, D.: Solving Random Satisfiable 3CNF Formulas in Expected Polynomial Time. In: Proceedings of 17th ACM-SIAM Symp. on Discrete Algorithms (SODA 2006), pp. 454–463 (2006)
McSherry, F.: Spectral Partitioning of Random Graphs. In: Proceedings of 42nd Annual IEEE Symp. on Foundations of Computer Science (FOCS 2001), pp. 529–523 (2001)
Scott, A.D., Sorkin, G.B.: Faster Algorithms for MAX CUT and MAX CSP, with Polynomial Expected Time for Sparse Instances. In: Proceedings of APPROX-RANDOM 2003, pp. 382–395 (2003)
Watanabe, O., Yamamoto, M.: Average-case Analysis for the MAX-2SAT Problem. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 277–282. Springer, Heidelberg (2006)
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Yamamoto, M. (2007). A Spectral Method for MAX2SAT in the Planted Solution Model. In: Tokuyama, T. (eds) Algorithms and Computation. ISAAC 2007. Lecture Notes in Computer Science, vol 4835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77120-3_12
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DOI: https://doi.org/10.1007/978-3-540-77120-3_12
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