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Probabilistic Analysis of Graph Algorithms

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Computational Graph Theory

Part of the book series: Computing Supplementum ((COMPUTING,volume 7))

Abstract

Probabilistic Analysis of Graph Algorithms. We review some of the known results on the average case performance of graph algorithms. The analysis assumes that the problem instances are randomly selected from some reasonable distribution of problems. We consider two types of problem. The first sort is polynomially solvable in the worst case but there are algorithms with better average case performance. In particular we consider the all-pairs shortest path problem, the minimum spanning tree problem, the assignment problem and the cardinality matching problem in sparse graphs. Our second category of problems consists of problems which seem hard in the worst-case but still have algorithms with good average case performance. In particular we consider three NP-Complete problems; the Hamilton cycle problem, the graph bisection problem and graph colouring. In addition we consider the graph isomorphism problem whose exact complexity is still undetermined.

AMS Subject Classifications: 68Q25, 05C8O.

Zusammenfassung

Probabilistische Analyse von Graphenalgorithmen. Die Arbeit bietet einen Überblick über Ergebnisse zur durchschnittlichen Leistungsfähigkeit von Graphenalogrithmen, wobei stets vorausgesetzt wird, daß die Problembeispiele zufällig gemäß einer ‘vemünftigen’ Wahrscheinlichkeitsverteilung gewählt werden. Wir betrachten zwei Problemtypen. Der erste Typ ist polynomial lösbar im schlechtesten Fall, jedoch existieren nicht-polynomiale Lösungsalgorithmen mit besserer durchschnittlicher Leistungs-fähigkeit. Insbesondere betrachten wir das Problem der kürzesten Wege zwischen allen Knotenpaaren, das Problem der Minimalbäume, das Zuordnungsproblem und das ungewichtete Matchingproblem in dünn belegten Graphen. Der zweite Problemtyp besteht aus Problemen, die im schlechtesten Fall äuBerst schwer zu losen erscheinen, wofür es aber weiterhin Lösungsalgorithmen mit guter durch-schnittlicher.Leistungsfähigkeit gibt. Insbesondere betrachten wir drei NP-vollständige Probleme: das Problem der Hamiltonkreise, das Bisektionsproblem für Graphen und das Färbungsproblem. Darüber hinaus wird das Graphenisomorphie-problem behandelt, dessen exakte Komplexität noch unbestimmt ist.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    A. M. Frieze

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  1. A. M. Frieze
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Editor information

Editors and Affiliations

  1. Institut für Mathematik, Technische Universität München, Federal Republic of Germany

    G. Tinhofer

  2. Department of Computer Science, Stanford University, Calif., USA

    E. Mayr

  3. Institut für Informatik, Universität Würzburg, Federal Republic of Germany

    H. Noltemeier

  4. Institute of Computer Science, University of Wrocław, Poland

    M. M. Syslo

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© 1990 Springer-Verlag

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Frieze, A.M. (1990). Probabilistic Analysis of Graph Algorithms. In: Tinhofer, G., Mayr, E., Noltemeier, H., Syslo, M.M. (eds) Computational Graph Theory. Computing Supplementum, vol 7. Springer, Vienna. https://doi.org/10.1007/978-3-7091-9076-0_11

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  • DOI: https://doi.org/10.1007/978-3-7091-9076-0_11

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82177-0

  • Online ISBN: 978-3-7091-9076-0

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