No truthful mechanism can be better than n approximate for two natural problems

This work gives the first natural non-utilitarian problems for which the trivial napproxima-tion via VCG mechanisms is the best possible. That is, no truthful mechanism can be better than n approximate, where n is the number of agents. The problems we study are the min-max variant of the shortest path and the (directed) minimum spanning tree mechanism design problems. In these procurement auctions, agents own the edges of a network, and the corresponding edge costs are private. Instead of the total weight of the subnetwork, in the min-max variant we aim to minimize the maximum agent cost.