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Approximability of Latin Square Completion-Type Puzzles

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Book cover Fun with Algorithms (FUN 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8496))

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Abstract

Among many variations of pencil puzzles, Latin square Completion-Type puzzles (LSCP), such as Sudoku, Futoshiki and BlockSum, are quite popular for puzzle fans. Concerning these puzzles, the solvability has been investigated from the viewpoint of time complexity in the last decade; it has been shown that, in most of these puzzles, it is NP-complete to determine whether a given puzzle instance has a proper solution. In this paper, we investigate the approximability of LSCP. We formulate LSCP as the maximization problem that asks to fill as many cells as possible, under the Latin square condition and the inherent condition. We then propose simple generic approximation algorithms for LSCP and analyze their approximation ratios.

This work is partially supported by JSPS KAKENHI Grant Number 24106004, 25104521 and 25870661.

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References

  1. Andersson, D.: Hashiwokakero is NP-complete. Information Processing Letters 109(19), 1145–1146 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Béjar, R., Fernández, C., Mateu, C., Valls, M.: The Sudoku completion problem with rectangular hole pattern is NP-complete. Discrete Mathematics 312(22), 3306–3315 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Colbourn, C.J.: The complexity of completing partial latin squares. Discrete Applied Mathematics 8(1), 25–30 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cygan, M.: Improved approximation for 3-dimensional matching via bounded pathwidth local search. arXiv preprint arXiv:1304.1424 (2013)

    Google Scholar 

  5. Demaine, E.D., Okamoto, Y., Uehara, R., Uno, Y.: Computational complexity and an integer programming model of Shakashaka. In: CCCG, pp. 31–36 (2013)

    Google Scholar 

  6. Easton, T., Parker, R.G.: On completing latin squares. Discrete Applied Mathematics 113(2), 167–181 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hajirasouliha, I., Jowhari, H., Kumar, R., Sundaram, R.: On completing latin squares. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 524–535. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  8. Haraguchi, K.: The number of inequality signs in the design of Futoshiki puzzle. Journal of Information Processing 21(1), 26–32 (2013)

    Article  MathSciNet  Google Scholar 

  9. Haraguchi, K., Ono, H.: Blocksum is NP-complete. IEICE Transactions on Information and Systems 96(3), 481–488 (2013)

    Article  Google Scholar 

  10. Hearn, R.A., Demaine, E.D.: Games, puzzles, and computation. AK Peters, Limited (2009)

    Google Scholar 

  11. Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM Journal on Discrete Mathematics 2(1), 68–72 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103 (1972)

    Google Scholar 

  13. Kölker, J.: Kurodoko is NP-complete. Journal of Information Processing 20(3), 694–706 (2012)

    Article  Google Scholar 

  14. Kumar, S.R., Russell, A., Sundaram, R.: Approximating latin square extensions. Algorithmica 24(2), 128–138 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Miyamoto, T.: Black Belt KenKen: 300 Puzzles. Puzzlewright (2013)

    Google Scholar 

  16. Miyamoto, T.: Brown Belt KenKen: 300 Puzzles. Puzzlewright (2013)

    Google Scholar 

  17. Miyamoto, T.: Green Belt KenKen: 300 Puzzles. Puzzlewright (2013)

    Google Scholar 

  18. Miyamoto, T.: White Belt KenKen: 300 Puzzles. Puzzlewright (2013)

    Google Scholar 

  19. Yato, T., Seta, T.: Complexity and completeness of finding another solution and its application to puzzles. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences 86(5), 1052–1060 (2003)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Commerce, Otaru University of Commerce, Japan

    Kazuya Haraguchi

  2. Faculty of Economics, Kyushu University, Japan

    Hirotaka Ono

Authors
  1. Kazuya Haraguchi
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  2. Hirotaka Ono
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Editor information

Editors and Affiliations

  1. Dipartimento di Matematica e Informatica, Città Universitaria, Università degli Studi di Catania, Viale A. Doria, 6, 95125, Catania, Italy

    Alfredo Ferro

  2. Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, 56127, Pisa, Italy

    Fabrizio Luccio

  3. Institute of Theoretical Computer Science, ETH Zürich, Universitätsstrasse 6, 8092, Zürich, Switzerland

    Peter Widmayer

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© 2014 Springer International Publishing Switzerland

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Haraguchi, K., Ono, H. (2014). Approximability of Latin Square Completion-Type Puzzles. In: Ferro, A., Luccio, F., Widmayer, P. (eds) Fun with Algorithms. FUN 2014. Lecture Notes in Computer Science, vol 8496. Springer, Cham. https://doi.org/10.1007/978-3-319-07890-8_19

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  • DOI: https://doi.org/10.1007/978-3-319-07890-8_19

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07889-2

  • Online ISBN: 978-3-319-07890-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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