Strategyproof cost-sharing mechanisms for set cover and facility location games

https://doi.org/10.1016/j.dss.2004.08.004Get rights and content

Abstract

Strategyproof cost-sharing mechanisms, lying in the core, that recover 1/α fraction of the cost, are presented for the set cover and facility location games: α=O(log n) for the former and 1:861 for the latter. Our mechanisms utilize approximation algorithms for these problems based on the method of dual-fitting.

Introduction

Achieving truth-revealing, also called strategyproofness or incentive compatibility, is fundamental to mechanism design. In a cost sharing mechanism, the goal is to distribute the cost of a shared resource among its users in such a way that revealing true utility is a dominant strategy of users. Other considerations include budget balance: that the users are not charged in excess of the incurring cost, at the same time recovering as much of the cost as possible. Ideally, one would like to recover all the cost, but this may not always be possible. Instead, the mechanism is required to recover a 1/α fraction of the cost, with α being a measure of how good the mechanism is.

In this paper, we consider cost functions that are defined as optimization problems. For instance, in the set cover (facility location) game, the cost is defined by the optimum solution to a set cover (facility location) problem. The set cover problem is a versatile optimization problem and can be used to model many situations. It is one of the fundamental problems in optimization and approximation algorithms. The facility location problem has been widely studied in Operations Research and also in the context of network design.

A recent paper of Nisan and Ronen [17] and one by Lehmann et al. [9] also considered cost functions defined as optimization problems, though in the setting of auctions, rather than cost sharing. Indeed, they even dealt with situations in which the underlying optimization problems were NP-hard, by resorting to methods from the field of approximation algorithms. Much work has been done on obtaining strategyproof cost sharing mechanisms—for instance for the spanning tree game [1], [5], [6], [13], [14]. Once again, the underlying optimization problems of some of the interesting games are NP-hard, and strategyproof cost allocation for several such games have been studied in Refs. [2], [3], [4], [8], [19], again using methods from approximation algorithms.

Most of the work related to cost sharing mechanisms in general concentrate on achieving the much harder task of group-strategyproofness (i.e., the agents have no incentive to lie even if collusions are allowed). Moulin and Shenker [16] showed that one way to get a group-strategyproof cost sharing mechanism is to construct a cross-monotone cost sharing method. They also showed that it is, in fact, the only way. This severely restricts the class of group-strategyproof mechanisms. In this paper, we show that relaxing this condition to individual strategyproofness results in simpler and better mechanisms.

Another important concept in the framework of cooperative game theory is that of a core. An allocation is in the core if it ensures that no subset of agents have an incentive to secede, i.e., no subset of the agents is charged more than the stand alone cost of serving that subset. Indeed, a budget balanced group-strategyproof mechanism is always in the core; the same is not true for a strategyproof mechanism. Hence, we impose this condition additionally. However, we consider a weaker version of core that ensures that no subset of users that actually obtain the shared resource have an incentive to secede, as opposed to the standard definition that ensures the same for all possible subsets.

In this paper, we obtain strategyproof cost allocations in the core that recover 1/α fraction of the cost, for two fundamental games whose underlying optimization problems are NP-hard, the set cover game and the facility location game. For the former, α=O(log n), and for the latter, α=1:861. For the latter game, this is made possible by new approximation algorithms for the underlying optimization problem using the technique of dual fitting [11]. In retrospect, the natural greedy algorithm for the set cover problem (see Ref. [20]) can also be analyzed using this technique. We utilize this viewpoint for handling the set cover game. The facility location game was studied in Ref. [4], [8], who left the open problem of obtaining a group-strategyproof mechanism based on a constant factor approximation algorithm. Our paper partially answers this question. We give a strategyproof mechanism. Subsequently, Pal and Tardos [18] gave a cross-monotonic cost-sharing method for the facility location problem that recovers one-third of the cost. This gives a group-strategyproof mechanism for the facility location game that recovers one-third of the cost. However, recently, Immorlica et al. [7] have shown that no cross-monotone cost sharing method for the facility location game can recover more than one-third of the cost. They also show a similar bound of 1=n for the set cover game. In light of these results, our paper shows that relaxing group-strategyproofness to strategyproofness indeed gives mechanisms that recover a larger fraction of the cost.

In fact, our technique seems to be quite general. Towards this, we show that our technique also extends to a game defined by a variation of the set cover problem, called the set multicover problem (under certain assumptions on the utilities). Our technique also speeds up the Moulin-Shenker mechanism for submodular cost functions, using the cost sharing method defined by Jain and Vazirani [11] by a factor of n.

In Section 2, we formally define the set cover game and give the mechanism. We also extend the mechanism to the set multicover game. In Section 3, we do the same for the facility location game. Section 4 shows how our technique yields essentially the same mechanism as that of Ref. [11] for submodular cost functions.

Section snippets

The set cover cost sharing game

Let N be a set of bidders. For each coalition SN the cost of providing a service to the bidders in S is C(S). How do you share C(N) among all the bidders?

Definition 1

Given the set of bids, b1, b2, …, bn a cost sharing mechanism computes

  • (1)

    the set of successful bidders, A, who are provided the service, and

  • (2)

    the amount charged to each bidder, x1, x2, …, xn.

An important consideration is the representation of the costs. A natural and interesting (to the computer scientist) case is when the cost is given by a

Facility location

Definition 6 Metric uncapacitated facility location

F is a set of facilities and C is a set of cities. Each facility i has an opening cost fi and the cost of connecting a facility i with a city j is cij . The connection costs satisfy the triangle inequality. The problem is to find a subset of facilities to open, IF, and a way to connect each city to an open facility, ϕ: CI such that the total cost of opening the facilities and connecting cities to open facilities is minimized.

In the cost sharing problem, the cities are the bidders. The

Submodular cost functions

In this section, we deviate from the previous model and assume that the cost function is given by an oracle. We consider those cost functions that follow the economies of scale:

Definition 7

A cost function is submodular if

  • (1)

    C(Ø)=0,

  • (2)

    for any S, TN, C(S)+C(T)≥C(ST)+C(S∩T).

Algorithm 3 Mechanism for facility location

The constraint 2 can also be replaced bySTN,iT,C(S+i)C(S)C(T+i)C(T).

Jain and Vazirani [11] give a primal-dual type algorithm (JV algorithm) to compute a cross-monotonic cost sharing method for submodular cost functions. This combined

Acknowledgments

We would like to thank Aranyak Mehta and Amin Saberi for useful discussions. We would also like to thank the anonymous referees for several useful comments.

Nikhil Devanur got his undergraduate degree in Computer Science and Engineering from the Indian Institute of Technology, Bombay, in 2001. He then joined the College of Computing at Georgia Tech. He was the College's Dean's Fellow during 2001–2002. His research has been focussed on efficient computation of market equilibria, and algorithmic aspects of game theory, arising in the context of the internet, and other huge networks.

In joint work with some of his co-authors, he gave the first

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Nikhil Devanur got his undergraduate degree in Computer Science and Engineering from the Indian Institute of Technology, Bombay, in 2001. He then joined the College of Computing at Georgia Tech. He was the College's Dean's Fellow during 2001–2002. His research has been focussed on efficient computation of market equilibria, and algorithmic aspects of game theory, arising in the context of the internet, and other huge networks.

In joint work with some of his co-authors, he gave the first polynomial time algorithm for computing market equilibrium when buyers have linear utility functions. He is expected to graduate with a PhD in 2006 under the guidance of Dr. Vijay Vazirani.

Milena Mihail holds a Dimploma in Electrical Engineering form the National Technical University of Athens and a PhD in Computer Science from Harvard University. She has been a member of the technical staff and a manager at Bell Communications Research and has been an Associate Professor at Georgia Tech since 1999.

Vijay Vazirani got his Bachelor's degree in Computer Science from MIT in 1979, and his PhD from U.C. Berkeley in 1983. He is currently a Professor of Computer Science at Georgia Tech. His research career has been centered around the design of algorithms, together with work on complexity theory, cryptography, coding theory, and game theory. During the first 10 years of his career, he made seminal contributions to the classical maximum matching problem which has historically played a central role in the development of the theory of algorithms. In the 1990s, he had much influence on the theory of approximation algorithms through work on several of its fundamental problems. In 2001, he published a book on this topic with Springer-Verlag. Over the last few years, he has been working in the nascent area of algorithmic game theory which addresses new game theoretic/computational issues arising from the Internet.

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