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 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. © 2009 Springer.
CITATION STYLE
Chebolu, P., Frieze, A., Melsted, P., & Sorkin, G. B. (2009). Average-case analyses of Vickrey costs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 5687 LNCS, pp. 434–447). https://doi.org/10.1007/978-3-642-03685-9_33
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