Efficient truthful mechanisms for the single‐source shortest paths tree problem

Abstract

AbstractLet a communication network be modeled by an undirected graph $G=(V,E)$ of n nodes and m edges, and assume that edges are controlled by selfish agents, which privately hold the length of each owned edge. In this paper we analyze the problem of designing a truthful mechanism for computing one of the most popular network topologies, i.e. the single‐source shortest paths tree. More precisely, we study several realistic scenarios, in which each agent can own either a single edge or multiple edges of G. In particular, for the single‐edge scenario, we show that in the utilitarian case the problem can be efficiently solved in $O(mn \log{}\alpha(m,n))$ time, while in a meaningful non‐utilitarian case, namely that in which agents' valuation functions depend only on the edge lengths, it can be solved in $O(m + n \log n)$ time. On the other hand, for the multiple‐edge scenario, in the utilitarian case we show an $O(mn+n^2 \log n)$ time truthful mechanism, while in the same non‐utilitarian case we provide an n‐approximate truthful mechanism which can be implemented in $O(m n \,\alpha(m,n))$ time. We also show that in the special case in which, for every agent, the owned edges are all incident to the same node, then this latter mechanism has an almost optimal $O(m\,\alpha(m,n))$ runtime. Copyright © 2007 John Wiley & Sons, Ltd.