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Average-Case Analyses of Vickrey Costs

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2009, RANDOM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5687))

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Abstract

We explore the average-case “Vickrey” cost of structures in three random settings: the Vickrey cost of a shortest path in a complete graph or digraph with random edge weights; the Vickrey cost of a minimum spanning tree (MST) in a complete graph with random edge weights; and the Vickrey cost of a perfect matching in a complete bipartite graph with random edge weights. In each case, in the large-size limit, the Vickrey cost is precisely 2 times the (non-Vickrey) minimum cost, but this is the result of case-specific calculations, with no general reason found for it to be true.

Separately, we consider the problem of sparsifying a complete graph with random edge weights so that all-pairs shortest paths are preserved approximately. The problem of sparsifying a given graph so that for every pair of vertices, the length of the shortest path in the sparsified graph is within some multiplicative factor and/or additive constant of the original distance has received substantial study in theoretical computer science. For the complete digraph \({\vec{K}_n}\) with random edge weights, we show that whp Θ(n ln n) edges are necessary and sufficient for a spanning subgraph to give good all-pairs shortest paths approximations.

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Author information

Authors and Affiliations

  1. University of Liverpool, Liverpool, L69 3BX, U.K.

    Prasad Chebolu

  2. Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Alan Frieze & Páll Melsted

  3. IBM Research, Yorktown Heights, NY, 10598, USA

    Gregory B. Sorkin

Authors
  1. Prasad Chebolu
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  2. Alan Frieze
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  3. Páll Melsted
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  4. Gregory B. Sorkin
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Editor information

Editors and Affiliations

  1. Dept. of Applied Math and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Irit Dinur

  2. Institute for Computer Science and Applied Mathematics, University of Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Technion, Computer Science Department, 32000, Haifa, Israel

    Joseph Naor

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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© 2009 Springer-Verlag Berlin Heidelberg

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Chebolu, P., Frieze, A., Melsted, P., Sorkin, G.B. (2009). Average-Case Analyses of Vickrey Costs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_33

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  • DOI: https://doi.org/10.1007/978-3-642-03685-9_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03684-2

  • Online ISBN: 978-3-642-03685-9

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