Strategyproof Approximation Mechanisms for Location on Networks

We consider the problem of locating a facility on a network, represented by a graph. A set of strategicagents have different ideal locations for the facility; the cost of an agent is the distance between its ideallocation and the facility. A mechanism maps the locations reported by the agents to the location of thefacility. Specifically, we are interested in social choice mechanisms that do not utilize payments. Wewish to design mechanisms that are strategyproof, in the sense that agents can never benefit by lying,or, even better, group strategyproof, in the sense that a coalition of agents cannot all benefit by lying.At the same time, our mechanisms must provide a small approximation ratio with respect to one of twooptimization targets: the social cost or the maximum cost.We give an almost complete characterization of the feasible truthful approximation ratio underboth target functions, deterministic and randomized mechanisms, and with respect to different networktopologies. Our main results are: We show that a simple randomized mechanism is group strategyproofand gives a (2 − 2/n)-approximation for the social cost, where n is the number of agents, when thenetwork is a circle (known as a ring in the case of computer networks); we design a novel “hybrid”strategyproof randomized mechanism that provides a tight approximation ratio of 3/2 for the maximumcost when the network is a circle; and we show that no randomized SP mechanism can provide an ap-proximation ratio better than 2 − o(1) to the maximum cost even when the network is a tree, therebymatching a trivial upper bound of two. ∗Microsoft Israel R&D; Center, 13 Shenkar Street, Herzeliya 46725, Israel, and Schools of Mathematics and Computer Science,Tel Aviv University, Tel Aviv, 69978, Israel, Email: [email protected]. Research supported in part by a USA Israeli BSF grant,by a grant from the Israel Science Foundation, by an ERC advanced grant and by the Hermann Minkowski Minerva Center forGeometry at Tel Aviv University.†Microsoft Israel R&D; Center, 13 Shenkar Street, Herzeliya 46725, Israel, and School of Business Administration and Centerfor the Study of Rationality, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Email: [email protected]‡Microsoft Israel R&D; Center, 13 Shenkar Street, Herzeliya 46725, Israel. Email: [email protected].§Microsoft Israel R&D; Center, 13 Shenkar Street, Herzeliya 46725, Israel, and Technion, IIT, Haifa 32000, Israel. Email:[email protected]