Exact and Approximate Truthful Mechanisms for the Shortest Paths Tree Problem

Let 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. In this paper we analyze the problem of designing a truthful mechanism for computing one of the most popular structures in communication networks, i.e., the single-source shortest paths tree. More precisely, we will study several realistic scenarios, in which each agent can own either a single or multiple edges of G. In particular, for the single-edge case, we will show that: (i) in the classic utilitarian case, the problem can be solved efficiently in O(mnlogα(m,n)) time, where ±(m,n) is the inverse of the Ackermanns function; (ii) in a meaningful non-utilitarian case, namely that in which agents valuation functions only depend on the edge lengths, the problem can be solved in O(m+nlogn) time. Conversely, for the multiple-edges case, we will show in the utilitarian case an O(mP+nPlogn) time truthful mechanism, where P=O(n) denotes the number of agents participating in the solution, while in the same non-utilitarian case we will prove a general lower bound to the approximation ratio that can be achieved by any truthful mechanism, by showing that no c-approximate mechanism can exist, for any fixed c < 5+√13 / 3+√13.