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Fog of Search Resolver for Minimum Remaining Values Strategic Colouring of Graph

  • Saajid AbuluaihEmail author
  • Azlinah MohamedEmail author
  • Muthukkaruppan AnnamalaiEmail author
  • Hiroyuki IidaEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 937)

Abstract

Minimum Remaining Values (MRV) is a popular strategy used along with Backtracking algorithm to solve Constraint Satisfaction Problems such as the Graph Colouring Problem. A common issue with MRV is getting stuck on search plateaus when two or more variables have the same minimum remaining values. MRV breaks the tie by arbitrarily selecting one of them, which might turn out to be not the best choice to expand the search. The paper relates the cause of search plateaus in MRV to ‘Fog of Search’ (FoS), and consequently proposes improvements to MRV to resolve the situation. The improved MRV+ generates a secondary heuristics value called the Contribution Number, and employs it to resolve a FoS. The usefulness of the FoS resolver is illustrated on Sudoku puzzles, a good instance of Graph Colouring Problem. An extensive experiment involving ten thousand Sudoku puzzles classified under two difficulty categories (based on the Number of clues and the Distribution of the clues) and five difficulty levels (ranging from Extremely Easy to Evil puzzles) were conducted. The results show that the FoS resolver that implements MRV+ is able to limit the FoS situations to a minimal, and consequently drastically reduce the number of recursive calls and backtracking moves that are normally ensued in MRV.

Keywords

Fog of Search Search plateau Constraint satisfaction problem Graph colouring problem Minimum remaining values Contribution number Sudoku puzzles 

References

  1. 1.
    Poole, D.L., Mackworth, A.K.: Artificial Intelligence: Foundations of Computational Agents Artificial. Cambridge University Press (2010)Google Scholar
  2. 2.
    Edelkamp, S., Schrodl, S.: Heuristic Search: Theory and Applications. Morgan Kaufmann Publishers Inc. (2011)Google Scholar
  3. 3.
    Habbas, Z., Herrmann, F., Singer, D., Krajecki, M.: A methodological approach to implement CSP on FPGA. In: IEEE International Workshop on Rapid System Prototyping Shortening Path from Specification to Prototype (1999).  https://doi.org/10.1109/iwrsp.1999.779033
  4. 4.
    Russell, S., Norvig, P.: Artificial Intelligence A: Modern Approach, 3rd edn. Pearson (2010)Google Scholar
  5. 5.
    Sudo, Y., Kurihara, M., Yanagida, T.: Keeping the stability of solutions to dynamic fuzzy CSPs. In: IEEE International Conference on Systems, Man and Cybernetics, pp. 1002–1007 (2008)Google Scholar
  6. 6.
    Haralick, R.M., Shapiro, L.G.: The consistent labeling problem: Part I. IEEE Trans. Pattern Anal. Mach. Intell. 173–184 (1979).  https://doi.org/10.1109/tpami.1979.4766903
  7. 7.
    Jilg, J., Carter, J.: Sudoku evolution. In: 2009 International IEEE Consumer Electronics Society’s Games Innovations Conference, pp. 173–185 (2009).  https://doi.org/10.1109/icegic.2009.5293614
  8. 8.
    Mcguire, G., Tugemann, B., Civario, G.: There is no 16-clue sudoku: solving the sudoku minimum number of clues problem via hitting set enumeration. Exp. Math. 23, 190–217 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jiang, B., Xue, Y., Li, Y., Yan, G.: Sudoku puzzles generating: from easy to evil. Chin. J. Math. Pract. Theory 39, 1–7 (2009)Google Scholar
  10. 10.
    Kiesling, E.C.: On war without the fog. Mil. Rev. 85–87 (2001)Google Scholar
  11. 11.
    Shapiro, M.J.: The fog of war. Secur. Dialogue 36, 233–246 (2005).  https://doi.org/10.1177/0967010605054651CrossRefGoogle Scholar
  12. 12.
    Asai, M., Fukunaga, A.: Exploration among and within plateaus in greedy best-first search. In: International Conference on Automated Planning Schedule, pp. 11–19 (2017)Google Scholar
  13. 13.
    Abuluaih, S., Mohamed, A.H., Annamalai, M., Iida, H.: Reordering variables using contribution number strategy to neutralize sudoku sets. In: International Conference on Agents Artificial Intelligence, pp. 325–333 (2015).  https://doi.org/10.5220/0005188803250333
  14. 14.
    Norvig, P.: Solving Every Sudoku Puzzle (2010). http://www.norvig.com/sudoku.html
  15. 15.
    Lee, W.: Programming Sudoku, 1st edn. Apress (2006)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Faculty of Computer and Mathematical SciencesUniversiti Teknologi MARAShah AlamMalaysia
  2. 2.Faculty of Computer and Mathematical Sciences, Advanced Analytic Engineering Center (AAEC)Universiti Teknologi MARAShah AlamMalaysia
  3. 3.School of Information ScienceJapan Advanced Institute of Science and Technology (JAIST)IshikawaJapan

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