Skip to main content
SpringerLink
Log in

Medians of Permutations: Building Constraints

  • Conference paper
Algorithms and Discrete Applied Mathematics (CALDAM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9602))

Included in the following conference series:

Abstract

Given a set \(\mathcal {A}\subseteq {\mathcal S}_n\) of m permutations of [n] and a distance function d, the median problem consists of finding a permutation \(\pi ^*\) that is the “closest” of the m given permutations. Here, we study the problem under the Kendall-\(\tau \) distance which counts the number of pairwise disagreements between permutations. This problem has been proved to be NP-hard when \(m \ge 4\), m even. In this article, we investigate new theoretical properties of \(\mathcal {A}\) that will solve the relative order between pairs of elements in median permutations of \(\mathcal {A}\), thus drastically reducing the search space of the problem.

supported by NSERC through an Individual Discovery Grant (Hamel) and by FRQNT through a Master’s scholarship (Milosz).

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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.

    Proof available here: http://www-etud.iro.umontreal.ca/miloszro/caldam/caldam.html

  2. 2.

    http://www-etud.iro.umontreal.ca/miloszro/caldam/caldam.html.

References

  1. Ailon, N., Charikar, M., Newman, N.: Aggregating inconsistent information: ranking and clustering. J. ACM 55(5), 1–27 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Betzler, N., Bredereck, R., Niedermeier, R.: Theoretical and empirical evaluation of data reduction for exact kemeny rank aggregation. Auton. Agent. Multi-Agent Syst. 28, 721–748 (2014)

    Article  Google Scholar 

  3. Betzler, N., et al.: Average parameterization and partial kernelization for computing medians. J. Comput. Syst. Sci. 77(4), 774–789 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Biedl, T.C., Brandenburg, F.J., Deng, X.: Crossings and permutations. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 1–12. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  5. Blin, G., Crochemore, M., Hamel, S., Vialette, S.: Median of an odd number of permutations. Pure Math. Appl. 21(2), 161–175 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Dwork, C., Kumar, R., Naor, M., Sivakumar, D.: Rank aggregation methods for the web. In: Proceedings of the 10th WWW, pp. 613–622 (2001)

    Google Scholar 

  7. Fisher, R.A., Yates, F.: Statistical Tables for Biological, Agricultural, Medical Research, 3rd edn., pp. 26–27. Oliver & Boyd, London (1948)

    Google Scholar 

  8. Karpinski, M., Schudy, W.: Faster algorithms for feedback arc set tournament, kemeny rank aggregation and betweenness tournament. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 3–14. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  9. Kemeny, J.: Mathematics without numbers. Daedalus 88, 577591 (1959)

    Google Scholar 

  10. Kendall, M.: A new measure of rank correlation. Biometrika 30, 81–89 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kenyon-Mathieu, C., Schudy, W.: How to rank with few errors, STOC 2007, pp. 95–103 (2007)

    Google Scholar 

  12. Knuth, D.E.: Seminumerical algorithms, The Art of Computer Programming 2. Reading, MA, : AddisonWesley, pp. 124–125 (1969)

    Google Scholar 

  13. Nishimura, N., Simjour, N.: Parameterized enumeration of (locally-) optimal aggregations. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 512–523. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  14. Simjour, N.: Improved parameterized algorithms for the kemeny aggregation problem. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 312–323. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  15. Truchon, M.: An Extension of the Condorcet Criterion and Kemeny Orders. Internal report, Université Laval, p. 16 (1998)

    Google Scholar 

  16. vanZuylen, A., Williamson, D.P.: Deterministic pivoting algorithms for constrained ranking and clustering problems. Math. Oper. Res. 34(3), 594–620 (2009)

    Google Scholar 

Download references

Acknowledgements

Thanks to Bryan Brancotte, Sarah Cohen-Boulakia and Alain Denise (LRI - Paris Sud) for giving us useful advices and thoughts to guide the work. Thanks to Nicole Burke (Montreal) for a careful english revision of the article.

Author information

Authors and Affiliations

  1. DIRO - Université de Montréal, C. P. 6128 Succ. Centre-Ville, Montréal, QC, H3C 3J7, Canada

    Robin Milosz & Sylvie Hamel

Authors
  1. Robin Milosz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sylvie Hamel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sylvie Hamel .

Editor information

Editors and Affiliations

  1. Indian Institute of Science, Bangalore, India

    Sathish Govindarajan

  2. Carleton University, Ottawa, ON, Canada

    Anil Maheshwari

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Milosz, R., Hamel, S. (2016). Medians of Permutations: Building Constraints. In: Govindarajan, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2016. Lecture Notes in Computer Science(), vol 9602. Springer, Cham. https://doi.org/10.1007/978-3-319-29221-2_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-29221-2_23

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29220-5

  • Online ISBN: 978-3-319-29221-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation