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Abstract

The quest for optimal/stable paths in graphs concerns a few practical or theoretical areas. Taking part in the quest, this paper adopts an abstract, general, equilibrium-oriented approach: it uses (quasi-arbitrary) arc-labelled digraphs, and assumes little about the structure of the sought paths and the definition of equilibrium, i.e. optimality/stability. The paper gives both a sufficient condition and a necessary condition for equilibrium existence for every “graph”, pinpoints the difference between these conditions, and shows coincidence when optimality relates to a total order. These results are applied to network routing.

Keywords

Labelled directed graph path preference equilibrium optimisation strict weak order sufficient condition necessary condition induction 

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References

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    Fishburn, P.: Intransitive indifference in preference theory: A survey. Operations Research 18, 207–228 (1970)zbMATHMathSciNetCrossRefGoogle Scholar
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    Griffin, T., Sobrinho, J.: Metarouting. In: SIGCOMM 2005: Proceedings of the 2005 conference on Applications, technologies, architectures, and protocols for computer communications, pp. 1–12. ACM Press, New York (2005)CrossRefGoogle Scholar
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    Lawler, E.: Combinatorial Optimization: Networks and Matroids. Dover (2001)Google Scholar
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    Le Roux, S.: Graphs and path equilibria. Research report, INRIA (2007), http://hal.inria.fr/inria-00195379/fr/

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Stéphane Le Roux
    • 1
  1. 1.École normale supérieure de LyonUniversité de Lyon, LIP, CNRS, INRIA, UCBL 

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