Skip to main content
SpringerLink
Log in

A Connection Between Sports and Matroids: How Many Teams Can We Beat?

  • Published:
Algorithmica Aims and scope Submit manuscript

A Correction to this article was published on 04 October 2017

This article has been updated

Abstract

Given an on-going sports competition, with each team having a current score and some matches left to be played, we ask whether it is possible for our distinguished team t to obtain a final standing with at most r teams finishing before t. We study the computational complexity of this problem, addressing it both from the viewpoint of parameterized complexity and of approximation. We focus on a special case equivalent to finding a maximal induced subgraph of a given graph G that admits an orientation where the in-degree of each vertex is upper-bounded by a given function. We obtain a \(\varTheta (\log |V(G)|)\) approximation for this problem based on an asymptotically optimal approximation we present for a certain matroid problem in which we need to cover a base of a matroid by picking elements from a set family.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

  • 04 October 2017

    The authors regret the following error in their article “A connection between sports and matroids: How many teams can we beat?” (Algorithmica, doi: 10.1007/s00453-016-0256-2), considering the computational complexity of the problem MinStanding(S). In Theorem 3 of our paper [4], we erroneously claimed a $$\mathsf {W}[1]$$ W [ 1 ] -hardness result to hold even for the case where the undirected version of the input graph is claw-free. In fact, we can only prove the theorem for $$K_{1,4}$$ K 1 , 4 -free graphs, so the property claw-free should have been replaced by the property $$K_{1,4}$$ K 1 , 4 -free.

Notes

  1. Actually we need certain small technical assumptions on S, essentialy to rule out the degenerate case when the matches can only have one possible outcome.

  2. In fact, this situation occurs at latest when each team has at most one remaining match to be played.

  3. In fact, Richey and Punnen [27] proved \(\mathsf {NP}\)-hardness for a slightly more general version of this problem, but their reduction proves the stronger result stated here.

References

  1. Adler, I., Erera, A.L., Hochbaum, D.S., Olinick, E.V.: Baseball, optimization and the world wide web. Interfaces 32(2), 12–22 (2002)

    Article  Google Scholar 

  2. Asahiro, Y., Jansson, J., Miyano, E., Ono, H.: Upper and lower degree bounded graph orientation with minimum penalty. In: CATS 2012: Proceedings of the 18th Computing: The Australasian Theory Symposium, vol. 128 of Conferences in Research and Practice in Information Technology, pp. 139–146. (2012)

  3. Bernholt, T., Gülich, A., Hofmeister, T., Schmitt, N.: Football elimination is hard to decide under the 3-point-rule. In: MFCS 1999: Proceedings of the 24th International Symposium on Mathematical Foundations of Computer Science, volume 1672 of Lecture Notes in Computer Science, pp. 410–418. Springer, (1999)

  4. Cechlárová, K., Potpinková, E., Schlotter, I.: Refining the complexity of the sports elimination problem. Discrete Appl. Math. 199, 172–186 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chrobak, M., Eppstein, D.: Planar orientations with low out-degree and compaction of adjacency matrices. Theor. Comput. Sci. 86, 243–266 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diestel, R.: Graph Theory, Volume 173 of Graduate Texts in Mathematics, vol. 173. Springer-Verlag, Berlin (2005)

    Google Scholar 

  7. Disser, Y., Matuschke, J.: Degree-constrained orientations of embedded graphs. J. Comb. Opt. 31(2), 758–773 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London (2013)

    Book  MATH  Google Scholar 

  9. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer-Verlag, New York (1999)

    Book  Google Scholar 

  10. Felsner, S.: Lattice structures from planar graphs. Electron. J. Comb. 11, R15 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Frank, A., Gyárfás, A.: How to orient the edges of a graph. Colloq. Math. Soc. János Bolyai 18, 353–364 (1976)

    MathSciNet  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. A Series of Books in the Mathematical Sciences. W. H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  13. Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gusfield, D., Martel, C.U.: A fast algorithm for the generalized parametric minimum cut problem and applications. Algorithmica 7(5&6), 499–519 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gusfield, D., Martel, C.U.: The structure and complexity of sports elimination numbers. Algorithmica 32(1), 73–86 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hakimi, S.L.: On the degrees of the vertices of a directed graph. J. Franklin Inst. 279, 290–308 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hermelin, D., Mnich, M., Leeuwen, E.J.: Parameterized complexity of induced \(H\)-matching on claw-free graphs. In: ESA 2012: Proceedings of the 16th Annual European Symposium on Algorithms, volume 7501 of Lecture Notes in Computer Science, pp. 624–635. Springer (2012)

  18. Hoffman, A.J., Rivlin, T.J.: When is a team “mathematically” eliminated?. In: Proceedings of the Princeton Symposium on Mathematical Programming, pp. 391–401. Princeton University Press (1970)

  19. Kern, W., Paulusma, D.: The new fifa rules are hard: complexity aspects of sports competitions. Discrete Appl. Math. 108(3), 317–323 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kern, W., Paulusma, D.: The computational complexity of the elimination problem in generalized sports competitions. Discrete Opt. 1(2), 205–214 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lawler, E.L.: Combinatorial optimization: networks and matroids. Dover Books on Mathematics, Courier Corporation (2012)

  22. McCormick, S.T.: Fast algorithms for parametric scheduling come from extensions to parametric maximum flow. Op. Res. 47(5), 744–756 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moshkovitz, D.: The projection games conjecture and the NP-hardness of \(\ln n\)-approximating Set-Cover. Theor. Comput. 11, 221–235 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Neumann, S., Wiese, A.: This house proves that debating is harder than soccer. In: FUN 2016: Proceedings of the 8th International Conference on Fun with Algorithms, pp. 25:1–25:14 (2016)

  25. Oxley, J.G.: Matroid Theory. Oxford graduate texts in mathematics. Oxford University Press, Oxford (2006)

    Google Scholar 

  26. Perfect, H.: Applications of menger’s graph theorem. J. Math. Anal. Appl. 22, 96–110 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  27. Richey, M.B., Punnen, A.P.: Minimum perfect bipartite matchings and spanning trees under categorization. Discrete Appl. Math. 39, 147–153 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Robinson, L.W.: Baseball playoff eliminations: an application of linear programming. Op. Res. Lett. 10(2), 67–74 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schwartz, B.L.: Possible winners in partially completed tournaments. SIAM Rev. 8, 302–308 (1966)

    Article  MATH  Google Scholar 

  30. Wayne, K.D.: A new property and a faster algorithm for baseball elimination. SIAM J. Discrete Math. 14(2), 223–229 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Budapest University of Technology and Economics, Budapest, 1521, Hungary

    Ildikó Schlotter

  2. Institute of Mathematics, Faculty of Science, P.J. Šafárik University, Jesenná 5, 040 01, Košice, Slovakia

    Katarína Cechlárová

Authors
  1. Ildikó Schlotter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Katarína Cechlárová
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ildikó Schlotter.

Additional information

I. Schlotter: Supported by the Hungarian Scientific Research Fund (OTKA Grants Nos. K-108383 and K-108947).

K. Cechlárová: Supported by the grants VEGA 1/0344/14, VEGA 1/0142/15, and APVV-15-0091.

A correction to this article is available online at https://doi.org/10.1007/s00453-017-0378-1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schlotter, I., Cechlárová, K. A Connection Between Sports and Matroids: How Many Teams Can We Beat?. Algorithmica 80, 258–278 (2018). https://doi.org/10.1007/s00453-016-0256-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-016-0256-2

Keywords

Search

Navigation