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Thresholds for Sports Elimination Numbers: Algorithms and Complexity

  • Dan Gusfield
  • Chip Martel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1663)

Abstract

Identifying teams eliminated from contention for first place of a sports league is a much studied problem. In the classic setting each game is played between two teams, and the team with the most wins finishes first. Recently, two papers [Way] and [AEHO] detailed a surprising structural fact in the classic setting: At any point in the season, there is a computable threshold W such that a team is eliminated (cannot win or tie for first place) if and only if it cannot win W or more games. Using this threshold speeds up the identification of eliminated teams.

We show that thresholds exist for a wide range of elimination problems (greatly generalizing the classical setting), via a simpler and more direct proof. For the classic setting we determine which teams can be the strict winner of the most games; examine these issues for multi-division leagues with playoffs and wildcards; and establish that certain elimination questions are NP-hard.

Keywords

Classic Setting Sport League Elimination Problem Threshold Result Single Team 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AEHO]
    I. Adler, A. Erera, D. Hochbaum, and E. Olinich. Baseball, optimization and the world wide web. unpublished manuscript 1998.Google Scholar
  2. [BGHS]
    T. Burnholt, A. Gullich, T. Hofmeister, and N. Schmitt. Football elimination is hard to decide under the 3-point rule. unpublished manuscript, 1999.Google Scholar
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    D. Gusfield and C. Martel. A fast algorithm for the generalized parametric minimum cut problem and applications. Algorithmica, 7:499–519, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
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    D. Gusfield and C. Martel. The structure and complexity of sports elimination numbers. Technical Report CSE-99-1, http://www.theory.cs.ucdavis.edu/, Univ. of California, Davis, 1999.Google Scholar
  5. [HR70]
    A. Hoffman and J. Rivlin. When is a team “mathematically” eliminated? In H.W. Kuhn, editor, Princeton symposium on math programming (1967), pages 391–401. Princeton univ. press, 1970.Google Scholar
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    T. McCormick. Fast algorithms for parametric scheduling come from extensions to parametric maximum flow. Proc. of 28th ACM Symposium on the Theory of Computing, pages 394–422, 1996.Google Scholar
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    B. Schwartz. Possible winners in partially completed tournaments. SIAM Review, 8: 302–308, 1966.zbMATHCrossRefGoogle Scholar
  8. [Way]
    K. Wayne. A new property and a faster algorithm for baseball elimination. ACM/SIAM Symposium on Discrete Algorithms, Jan. 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Dan Gusfield
    • 1
  • Chip Martel
    • 2
  1. 1.Department of Computer ScienceUniversity of California, DavisDavis
  2. 2.Department of Computer ScienceUniversity of California, DavisDavis

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