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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.

Keywords

Average-case analysis VCG auction random graph shortest path minimum spanning tree MST Random Assignment Problem 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Prasad Chebolu
    • 1
  • Alan Frieze
    • 2
  • Páll Melsted
    • 2
  • Gregory B. Sorkin
    • 3
  1. 1.University of LiverpoolLiverpoolU.K.
  2. 2.Carnegie Mellon UniversityPittsburghUSA
  3. 3.IBM ResearchNYUSA

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