True costs of cheap labor are hard to measure: edge deletion and VCG payments in graphs

We address the problem of lowering the buyer's expected payments in shortest path auctions, where the buyer's goal is to purchase a path in a graph in which edges are owned by selfish agents. We show that by deleting some of the edges of the graph, one can reduce the total payment of the VCG mechanism by a factor of θ(n). However, we prove that it is NP-hard to find the best subset of edges to delete, even if the edge costs are small integers, or the graph has very simple structure; in the former case, this problem is hard to approximate, too. On the positive side, we describe a pseudopolynomial time algorithm for series-parallel graphs and fixed edge costs. Also, we discuss the applicability of this algorithm for the case of general (probabilistic) costs and derive a general lower bound on the performance of algorithms that are based on expected edge costs.