Abstract
Nested Monte-Carlo Search (NMC) and Nested Rollout Policy Adaptation (NRPA) are Monte-Carlo tree search algorithms that have proved their efficiency at solving one-player game problems, such as morpion solitaire or sudoku 16x16, showing that these heuristics could potentially be applied to constraint problems. In the field of Ramsey theory, the weak Schur number WS(k) is the largest integer n for which their exists a partition into k subsets of the integers [1,n] such that there is no x < y < z all in the same subset with x + y = z. Several studies have tackled the search for better lower bounds for the Weak Schur numbers WS(k), k ≥ 4. In this paper we investigate this problem using NMC and NRPA, and obtain a new lower bound for WS(6), namely WS(6) ≥ 582.
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References
Abbott, H., Hanson, D.: A problem of Schur and its generalizations. Acta Arith. 20, 175–187 (1972)
Blanchard, P.F., Harary, F., Reis, R.: Partitions into sum-free sets. Integers 6 A7 (2006)
Bornsztein, P.: On an extension of a theorem of Schur. Acta Arith. 101, 395–399 (2002)
Brown, T., Landman, B.M., Robertson, A.: Note: Bounds on some van der waerden numbers. J. Comb. Theory Ser. A 115(7), 1304–1309 (2008)
Cazenave, T.: Nested Monte-Carlo search. In: Boutilier, C. (ed.) IJCAI, pp. 456–461 (2009)
Cazenave, T., Teytaud, F.: Application of the Nested Rollout Policy Adaptation Algorithm to the Traveling Salesman Problem with Time Windows. In: Hamadi, Y., Schoenauer, M. (eds.) LION 2012. LNCS, vol. 7219, pp. 42–54. Springer, Heidelberg (2012)
Drake, P.: The last-good-reply policy for Monte-Carlo go. ICGA Journal 32(4), 221–227 (2009)
Eliahou, S., Marín, J.M., Revuelta, M.P., Sanz, M.I.: Weak Schur numbers and the search for G. W. Walker’s lost partitions. Computer and Mathematics with Applications 63, 175–182 (2012)
Robilliard, D., Fonlupt, C., Marion-Poty, V., Boumaza, A.: A Multilevel Tabu Search with Backtracking for Exploring Weak Schur Numbers. In: Hao, J.-K., Legrand, P., Collet, P., Monmarché, N., Lutton, E., Schoenauer, M. (eds.) EA 2011. LNCS, vol. 7401, pp. 109–119. Springer, Heidelberg (2012)
Fredricksen, H., Sweet, M.M.: Symmetric sum-free partitions and lower bounds for Schur numbers. Electr. J. Comb. 7 (2000)
Gelly, S., Silver, D.: Combining online and offline knowledge in uct. In: Ghahramani, Z. (ed.) ICML. ACM International Conference Proceeding Series, vol. 227, pp. 273–280. ACM (2007)
Le Bras, R., Gomes, C.P., Selman, B.: From streamlined combinatorial search to efficient constructive procedures. In: Proceedings of the 15th International Conference on Artificial Intelligence, AAAI 2012 (2012)
Rado, R.: Some solved and unsolved problems in the theory of numbers. Math. Gaz. 25, 72–77 (1941)
Rimmel, A., Teytaud, F.: Multiple Overlapping Tiles for Contextual Monte Carlo Tree Search. In: Di Chio, C., Cagnoni, S., Cotta, C., Ebner, M., Ekárt, A., Esparcia-Alcazar, A.I., Goh, C.-K., Merelo, J.J., Neri, F., Preuss, M., Togelius, J., Yannakakis, G.N. (eds.) EvoApplicatons 2010, Part I. LNCS, vol. 6024, pp. 201–210. Springer, Heidelberg (2010)
Rimmel, A., Teytaud, F., Cazenave, T.: Optimization of the Nested Monte-Carlo Algorithm on the Traveling Salesman Problem with Time Windows. In: Di Chio, C., Brabazon, A., Di Caro, G.A., Drechsler, R., Farooq, M., Grahl, J., Greenfield, G., Prins, C., Romero, J., Squillero, G., Tarantino, E., Tettamanzi, A.G.B., Urquhart, N., Uyar, A.Ş. (eds.) EvoApplications 2011, Part II. LNCS, vol. 6625, pp. 501–510. Springer, Heidelberg (2011)
Rimmel, A., Teytaud, F., Teytaud, O.: Biasing Monte-Carlo Simulations through RAVE Values. In: van den Herik, H.J., Iida, H., Plaat, A. (eds.) CG 2010. LNCS, vol. 6515, pp. 59–68. Springer, Heidelberg (2011)
Rosin, C.D.: Nested rollout policy adaptation for Monte Carlo tree search. In: Walsh, T. (ed.) IJCAI, pp. 649–654. IJCAI/AAAI (2011)
Schur, I.: Über die kongruenz x m + y m ≡ z m (mod p). Jahresbericht der Deutschen Mathematiker Vereinigung 25, 114–117 (1916)
Walker, G.: A problem in partitioning. Amer. Math. Monthly 59, 253 (1952)
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Eliahou, S., Fonlupt, C., Fromentin, J., Marion-Poty, V., Robilliard, D., Teytaud, F. (2013). Investigating Monte-Carlo Methods on the Weak Schur Problem. In: Middendorf, M., Blum, C. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2013. Lecture Notes in Computer Science, vol 7832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37198-1_17
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DOI: https://doi.org/10.1007/978-3-642-37198-1_17
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