Investigating Monte-Carlo Methods on the Weak Schur Problem

  • Shalom Eliahou
  • Cyril Fonlupt
  • Jean Fromentin
  • Virginie Marion-Poty
  • Denis Robilliard
  • Fabien Teytaud
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7832)


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.


Expert Knowledge Tree Search Algorithm Ramsey Theory Good Lower Bound Terminal Word 
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  1. 1.
    Abbott, H., Hanson, D.: A problem of Schur and its generalizations. Acta Arith. 20, 175–187 (1972)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Blanchard, P.F., Harary, F., Reis, R.: Partitions into sum-free sets. Integers 6 A7 (2006)Google Scholar
  3. 3.
    Bornsztein, P.: On an extension of a theorem of Schur. Acta Arith. 101, 395–399 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cazenave, T.: Nested Monte-Carlo search. In: Boutilier, C. (ed.) IJCAI, pp. 456–461 (2009)Google Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Drake, P.: The last-good-reply policy for Monte-Carlo go. ICGA Journal 32(4), 221–227 (2009)MathSciNetGoogle Scholar
  8. 8.
    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)zbMATHCrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Fredricksen, H., Sweet, M.M.: Symmetric sum-free partitions and lower bounds for Schur numbers. Electr. J. Comb. 7 (2000)Google Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Rado, R.: Some solved and unsolved problems in the theory of numbers. Math. Gaz. 25, 72–77 (1941)CrossRefGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    Rosin, C.D.: Nested rollout policy adaptation for Monte Carlo tree search. In: Walsh, T. (ed.) IJCAI, pp. 649–654. IJCAI/AAAI (2011)Google Scholar
  18. 18.
    Schur, I.: Über die kongruenz x m + y m ≡ z m (mod p). Jahresbericht der Deutschen Mathematiker Vereinigung 25, 114–117 (1916)zbMATHGoogle Scholar
  19. 19.
    Walker, G.: A problem in partitioning. Amer. Math. Monthly 59, 253 (1952)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Shalom Eliahou
    • 1
  • Cyril Fonlupt
    • 2
  • Jean Fromentin
    • 1
  • Virginie Marion-Poty
    • 2
  • Denis Robilliard
    • 2
  • Fabien Teytaud
    • 2
  1. 1.Univ Lille Nord de France, ULCO, LISICCalaisFrance
  2. 2.Univ Lille Nord de France, ULCO, LMPACalaisFrance

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