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The polytope of win vectors

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Abstract

Imagine a graph as representing a fixture list with vertices corresponding to teams, and the number of edges joiningu andv as representing the number of games in whichu andv have to play each other. Each game ends in a win, loss, or tie and we say a vector\(\vec w\)=(w 1,...,w n) is awin vector if it represents the possible outcomes of the games, withw i denoting the total number of games won by teami. We study combinatorial and geometric properties of the set of win vectors and, in particular, we consider the problem of counting them. We construct a fully polynomial randomized approximation scheme for their number in dense graphs. To do this we prove that the convex hull of the set of win vectors ofG forms an integral polymatroid and then use volume approximation techniques.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 27-29 St Giles, OX1 3LB, Oxford, UK

    J. E. Bartels & D. J. A. Welsh

  2. Arris Pharmaceutical, 385 Oyster Pt., Blvd., Suite 3, 94080, South San Francisco, CA, USA

    J. Mount

Authors
  1. J. E. Bartels
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  2. J. Mount
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  3. D. J. A. Welsh
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Supported by the “DAAD Doktorandenstipendium des zweiten Hochschulsonderprogrammes HSPII/AUFE”.

Partially supported by RAND-REC EC US030.

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Bartels, J.E., Mount, J. & Welsh, D.J.A. The polytope of win vectors. Annals of Combinatorics 1, 1–15 (1997). https://doi.org/10.1007/BF02558460

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  • DOI: https://doi.org/10.1007/BF02558460

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