Skip to main content
SpringerLink
Log in

A Constructive Algorithm for Partial Latin Square Extension Problem that Solves Hardest Instances Effectively

  • Chapter
  • First Online:
Recent Advances in Computational Optimization

Part of the book series: Studies in Computational Intelligence ((SCI,volume 580))

Abstract

A partial Latin square (PLS) is a partial assignment of \(n\) symbols to an \(n\times n\) grid such that, in each row and in each column, each symbol appears at most once. The partial Latin square extension (PLSE) problem asks to find such a PLS that is a maximum extension of a given PLS. Recently Haraguchi et al. proposed a heuristic algorithm for the PLSE problem. In this paper, we present its effectiveness especially for the “hardest” instances. We show by empirical studies that, when \(n\) is large to some extent, the instances such that symbols are given in 60–70 % of the \(n^2\) cells are the hardest. For such instances, the algorithm delivers a better solution quickly than IBM ILOG CPLEX, a state-of-the-art optimization solver, that is given a longer time limit. It also outperforms surrogate constraint based heuristics that are originally developed for the maximum independent set problem.

This work is partially supported by JSPS KAKENHI Grant Number 25870661.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Gomes and Shmoys [3] claims that the boundary on whether the instance has an LS optimal solution or not should lie around \(r=0.42\).

  2. 2.

    The larger the extent of the neighborhood is, the smaller \(\varepsilon \) becomes.

References

  1. Colbourn, C.J.: The complexity of completing partial Latin squares. Discrete Appl. Math. 8, 25–30 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barry, R.A., Humblet, P.A.: Latin routers, design and implementation. IEEE/OSA J. Lightwave Technol. 11(5), 891–899 (1993)

    Article  Google Scholar 

  3. Gomes, C.P., Shmoys, D.: Completing quasigroups or Latin squares: a structured graph coloring problem. In: Proc. Computational Symposium on Graph Coloring and Generalizations (2002)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Haraguchi, K., Ishigaki, M., Maruoka, A.: A maximum matching based heuristic algorithm for partial latin square extension problem. In: Proc. FedCSIS 2013, 347–354 (2013)

    Google Scholar 

  6. IBM, ILOG CPLEX: http://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/

  7. Glover, F.: Tutorial on surrogate constraint approaches for optimization in graphs. J. Heuristics 9, 175–227 (2003)

    Article  MATH  Google Scholar 

  8. Alidaee, B., Kochenberger, G., Wang, H.: Simple and fast surrogate constraint heuristics for the maximum independent set problem. J. Heuristics 14, 571–585 (2008)

    Article  Google Scholar 

  9. Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cymer, R.: Dulmage-Mendelsohn canonical decomposition as a generic pruning technique. Constraints 17, 234–272 (2012)

    Article  MathSciNet  Google Scholar 

  11. Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman & Company, New York (1979)

    Google Scholar 

  12. Barták, R.: On generators of random quasigroup problems. In: Proc. CSCLP 2005, 164–178 (2006)

    Google Scholar 

  13. Gomes, G., Selman, B.: Problem structure in the presence of perturbations. In: Proc. AAAI 97, pp. 221–227 (1997)

    Google Scholar 

  14. Shaw, P., Stergiou, K., Walsh, T.: Arc consistency and quasigroup completion. In: Proc. ECAI-98 (workshop on binary constraints) (1998)

    Google Scholar 

  15. Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity from characteristic phase transitions. Nature 400, 133–137 (1999)

    Article  MathSciNet  Google Scholar 

  16. Ansotegui, C., Bejar, R., Fernandez, C., Mateu, C.: On the hardness of solving edge matching puzzles as sat or csp problems. Constraints 18, 7–37 (2013)

    Article  MathSciNet  Google Scholar 

  17. Appa, G., Magos, D., Mourtos, I.: Searching for mutually orthogonal Latin squares via integer and constraint programming. Eur. J. Oper. Res. 173(2), 519–530 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fontaine, D., Michel, L.: A high level language for solver independent model manipulation and generation of hybrid solvers. In: Proc. CPAIOR 2012, pp. 180–194 (2012)

    Google Scholar 

  19. Andrade, D., Resende, M., Werneck, R.: Fast local search for the maximum independent set problem. J. Heuristics 18, 525–547 (2012)

    Article  Google Scholar 

  20. LocalSolver: http://www.localsolver.com/

  21. Hajirasouliha, I., Jowhari, H., Kumar, R., Sundaram, R.: On completing Latin squares. In: Proc. STACS 2007, pp. 524–535

    Google Scholar 

  22. 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 J. Discrete Math. 2(1), 68–72 (1989)

    Article  MATH  MathSciNet  Google Scholar 

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

Download references

Author information

Authors and Affiliations

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

    Kazuya Haraguchi

Authors
  1. Kazuya Haraguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuya Haraguchi .

Editor information

Editors and Affiliations

  1. Department of Parallel Algorithms, Institute of Information and CommunicationTechnology, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Stefka Fidanova

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Haraguchi, K. (2015). A Constructive Algorithm for Partial Latin Square Extension Problem that Solves Hardest Instances Effectively. In: Fidanova, S. (eds) Recent Advances in Computational Optimization. Studies in Computational Intelligence, vol 580. Springer, Cham. https://doi.org/10.1007/978-3-319-12631-9_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-12631-9_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12630-2

  • Online ISBN: 978-3-319-12631-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation