On the Existence of Truthful Mechanisms for the Minimum-Cost Approximate Shortest-Paths Tree Problem

Abstract

Let a communication network be modeled by a graph G = (V,E) of n nodes and m edges, where with each edge is associated a pair of values, namely its cost and its length. Assume now that each edge is controlled by a selfish agent, which privately holds the cost of the edge. In this paper we analyze the problem of designing in this non-cooperative scenario a truthful mechanism for building a broadcasting tree aiming to balance costs and lengths. More precisely, given a root node r ∈V and a real value λ≥1, we want to find a minimum cost (as computed w.r.t. the edge costs) spanning tree of G rooted at r such that the maximum stretching factor on the distances from the root (as computed w.r.t. the edge lengths) is λ. We call such a tree the Minimum-cost λ -Approximate Shortest-paths Tree (λ-MAST).

First, we prove that, already for the unit length case, the λ-MAST problem is hard to approximate within better than a logarithmic factor, unless NP admits slightly superpolynomial time algorithms. After, assuming that the graph G is directed, we provide a (1 + ε)(n – 1)-approximate truthful mechanism for solving the problem, for any ε> 0. Finally, we analyze a variant of the problem in which the edge lengths coincide with the private costs, and we provide: (i) a constant lower bound (depending on λ) to the approximation ratio that can be achieved by any truthful mechanism; (ii) a $(1+ {{n-1}\over{\lambda}})$-approximate truthful mechanism.

Work partially supported by the Research Project GRID.IT, funded by the Italian Ministry of Education, University and Research, and by the European Union under IST FET Integrated Project 015964 AEOLUS and COST Action 293 GRAAL.