Randomized Truthful Mechanisms for Scheduling Unrelated Machines

In this work we study mechanisms with verification, as introduced by Nisan and Ronen [STOC 1999], to solve problems involving selfish agents. We provide a technique for designing truthful mechanisms with verification that optimally solve the underlying optimization problem. Problems (optimally) solved with our technique belong to a rich class that includes, as special cases, utilitarian problems and many others considered in literature for so called one-parameter agents (e.g., the make-span). Our technique extends the one recently presented by Auletta et al [ICALP 2006] as it works for any finite multi-dimensional valuation domain. As special case we obtain an alternative technique to optimally solve (though not in polynomial-time) Scheduling Unrelated Machines studied (and solved) by Nisan and Ronen. Interestingly enough, our technique also solves the case of compound agents (i.e., agents declaring more than a value). As an application we provide the first optimal truthful mechanism with verification for Scheduling Unrelated Machines in which every selfish agent controls more than one (unrelated) machine. We also provide methods leading to approximate solutions obtained with polynomial-time truthful mechanisms with verification. With such methods we obtain polynomial-time truthful mechanisms with verification for smooth problems involving compound agents composed by one-parameter valuations. Finally, we investigate the construction of mechanisms (with verification) for infinite domains. We show that existing techniques to obtain truthful mechanisms (for the case in which verification is not allowed), dealing with infinite domains, could completely annul advantages that verification implies.

We consider the problem of designing truthful mechanisms on m unrelated machines, to minimize some optimization goal. Nisan and Ronen [Nisan, N., A. Ronen. 2001. Algorithmic mechanism design. Games Econom. Behav. 35 166–196] consider the specific goal of makespan minimization, and show a lower bound of 2, and an upper bound of m. This large gap inspired many attempts that yielded positive results for several special cases, but very partial success for the general case: the lower bound was slightly increased to 2.61 by Christodoulou et al. [Christodoulou, G., E. Koutsoupias, A. Kovács. 2010. Mechanism design for fractional scheduling on unrelated machines. ACM Trans. Algorithms (TALG) 6(2) 1–18] and Koutsoupias and Vidali [Koutsoupias, E., A. Vidali. 2007. A lower bound of 1+phi for truthful scheduling mechanisms. Proc. 32nd Internat. Sympos. Math. Foundations Comput. Sci. (MFCS)], while the best upper bound remains unchanged. In this paper we show the optimal lower bound on truthful anonymous mechanisms: no such mechanism can guarantee an approximation ratio better than m. Moreover, our proof yields similar optimal bounds for two other optimization goals: the sum of completion times and the lp norm of the schedule.

We consider the design of strategyproof cost-sharing mechanisms. We give two simple, but extremely versatile, black-box reductions, that in combination reduce the cost-sharing mechanism-design problem to the algorithmic problem of finding a minimum-cost solution for a set of players. Our first reduction shows that any truthful, α-approximation mechanism for the social-cost minimization (SCM) problem satisfying a technical no-bossiness condition can be morphed into a truthful mechanism that achieves an O(α log n)-approximation where the prices recover the cost incurred. Thus, we decouple the task of truthfully computing an outcome with near-optimal social cost from the cost-sharing problem. This is fruitful since truthful mechanism-design, especially for single-dimensional problems, is a relatively well-understood and manageable task. Our second reduction nicely complements the first one by showing that any LP-based ρ-approximation for the problem of finding a min-cost solution for a set of players yields a truthful, no-bossy, (ρ + 1)-approximation for the SCM problem (and hence, a truthful (ρ + 1)log n-approximation cost-sharing mechanism).These reductions find a slew of applications, yielding, as corollaries, the first or improved polytime cost-sharing mechanisms for a variety of problems. For example, our first reduction coupled with the celebrated VCG mechanism shows that for any cost-sharing problem (with a monotone cost function) one can obtain a truthful mechanism that achieves an O(log n)-approximation where the prices recover the cost incurred. Other applications include O(log n)-approximation mechanisms for: survivable network design problems, facility location (FL) problems including capacitated and connected FL problems, and minimum-makespan scheduling on unrelated machines. Our results demonstrate that in contrast with our current understanding of group-strategyproof and acyclic mechanisms, strategyproofness allows for ample flexibility in cost-sharing mechanism design enabling one to effectively leverage various algorithmic results.

Differentiated service (DiffServ) is a mechanism to provide the Quality of Service (QoS) with a certain performance guarantee. In this paper, we study how to design DiffServ multicast when the participants (i.e., relay links) are selfish. We assume that each link e i is associated with a cost coefficient a i such that the cost of e i to provide a multicast service with bandwidth demand x is a i · x. We first show that a previous approximation algorithm does not directly induce a truthful mechanism. We then give a new polynomial time 8-approximation algorithm to construct a DiffServ multicast tree. Based on this tree, we design a truthful mechanism for DiffServ multicast, i.e., we give a polynomial-time computable payment scheme to compensate all chosen relay links such that each link e i maximizes its profit when it reports its privately cost coefficient a i truthfully.

In this paper, we focus on the facility location games with the property of dual preference. Dual preference property indicates that both two preferences of agents, staying close to and staying away from the facility(s), exist in the facility location game. We will explore two types of facility location games with this property, the dual character facility location game and the two-opposite-facility location game with limited distance which model the scenarios in real life. For both of them, we wish to design strategy proof mechanisms or group strategy-proof mechanisms with the objective of optimizing the social utility. For the dual character facility location game, we propose a strategy proof optimal mechanism when misreporting is restricted to agents' preferences, and give a 1/3-approximation deterministic group strategy-proof mechanism when both location and preference are considered as private information. For the two-opposite-facility location game with limited distance, when the number of agents is even (denoted as 2k), we give a 1/k-approximation deterministic group strategy-proof mechanism, and when the number of agents is odd (denoted as 2k-1), we propose a 1/(2k-1)-approximation deterministic group strategy-proof mechanism. The approximation ratios for both mechanisms are proved to be the best a deterministic strategy-proof mechanism can achieve.

Mechanisms for aggregating the preferences of agents in elections need to balance many different considerations, including efficiency, information elicited from agents, and manipulability. We consider the utilitarian social welfare of mechanisms for preference aggregation, measured by the distortion. We show that for a particular input format called threshold approval voting, where each agent is presented with an independently chosen threshold, there is a mechanism with nearly optimal distortion when the number of voters is large. Threshold mechanisms are potentially manipulable, but place a low informational burden on voters. We then consider truthful mechanisms. For the widely-studied class of ordinal mechanisms which elicit the rankings of candidates from each agent, we show that truthfulness essentially imposes no additional loss of welfare. We give truthful mechanisms with distortion O(√m log m) for k-winner elections, and distortion O(√m log m) when candidates have arbitrary costs, in elections with m candidates. These nearly match known lower bounds for ordinal mechanisms that ignore the strategic behavior. We further tighten these lower bounds and show that for truthful mechanisms our first upper bound is tight. Lastly, when agents decide between two candidates, we give tight bounds on the distortion for truthful mechanisms.

In this paper we address the question of designing truthful mechanisms for solving optimization problems on dynamic graphs. More precisely, we are given a graph G of n nodes, and we assume that each edge of G is owned by a selfish agent. The strategy of an agent consists in revealing to the system the cost for using its edge, but this cost is not constant and can change over time. Additionally, edges can enter into and exit from G. Among the various possible assumptions which can be made to model how these edge-cost modifications take place, we focus on two settings: (i) the dynamic, in which modifications are unpredictable and time-independent, and for a given optimization problem on G, the mechanism has to maintain efficiently the output specification and the payment scheme for the agents; (ii) the time-sequenced, in which modifications happens at fixed time steps, and the mechanism has to minimize an objective function which takes into consideration both the quality and the set-up cost of a new solution. In both settings, we investigate the existence of exact and approximate truthful mechanisms. In particular, for the dynamic setting, we analyze the minimum spanning tree problem, and we show that if edge costs can only decrease, then there exists an efficient dynamic truthful mechanism for handling a sequence of k edge-cost reductions having runtime , where h is the overall number of payment changes.

We study the problem of designing truthful mechanisms for makespan minimization in scheduling. In particular, we consider randomized mechanisms for a restriction of the general multi-dimensional domain (i.e., unrelated machines). In a sense, our setting is the simplest multi-dimensional setting, where each machine holds privately only a single-bit of information. Some of the impossibility results for deterministic mechanisms carry over our setting as well. We prove a separation between truthful-in-expectation and universally truthful mechanisms for makespan minimization: We first show how to design an optimal truthful-in-expectation mechanism, and then prove lower bounds on the approximation guarantee of universally truthful mechanisms.

Bounding the payment of approximate truthful mechanisms

We consider the problem of locating a facility on a network represented by a graph. A set of strategic agents have different ideal locations for the facility; the cost of an agent is the distance between its ideal location and the facility. A mechanism maps the locations reported by the agents to the location of the facility. We wish to design mechanisms that are strategyproof (SP) in the sense that agents can never benefit by lying and, at the same time, provide a small approximation ratio with respect to the minimax measure. We design a novel “hybrid” strategyproof randomized mechanism that provides a tight approximation ratio of 3/2 when the network is a circle (known as a ring in the case of computer networks). Furthermore, we show that no randomized SP mechanism can provide an approximation ratio better than 2 - o(1), even when the network is a tree, thereby matching a trivial upper bound of two.

Truthfulness with value-maximizing bidders: On the limits of approximation in combinatorial markets

Truthful mechanism for task allocation with lowest revenue limit in spatial crowdsourcing

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomized mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomized mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of eovere-1. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of 18 over 13 on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy, an improved lower bound of eovere-1 holds. This lower bound is tight given that RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomized strategy of the administrator who interviews the applicants.

Motivated by many practical applications, in this paper we study budget feasible mechanisms with the goal of procuring an independent set of a matroid. More specifically, we are given a matroid {mathcal {M}}=(E,{mathcal {I}}). Each element of the ground set E is controlled by a selfish agent and the cost of the element is private information of the agent itself. A budget limited buyer has additive valuations over the elements of E. The goal is to design an incentive compatible budget feasible mechanism which procures an independent set of the matroid of largest possible value. We also consider the more general case of the pair {mathcal {M}}=(E,{mathcal {I}}) satisfying only the hereditary property. This includes matroids as well as matroid intersection. We show that, given a polynomial time deterministic algorithm that returns an alpha -approximation to the problem of finding a maximum-value independent set in {mathcal {M}}, there exists an individually rational, truthful and budget feasible mechanism which is (3alpha +1)-approximated and runs in polynomial time, thus yielding also a 4-approximation for the special case of matroids.

The following problems are worth studying: Open problems on the Steiner ratio, such as Chung-Gilbert’s conjecture, Graham-Hwang’s conjecture, and Cielick’s conjecture, etc.. Find better approximation for network Steiner trees and establish an explicit lower bound for the approximation performance ratio of network Steiner trees. Close the gap between the lower bound and the upper bound for the approximation performance ratio of Steiner minimum arborescence. Find more efficient approximation algorithms for on-line and dynamic Steiner minimum trees and various Steiner tree packing problems. Find good approximation algorithms for multi-weighted Steiner trees and multiphase Steiner trees and study close relationship between multi-weighted Steiner trees, multiphase Steiner trees, and phylogenetic trees. Make clear whether there exists a polynomial-time approximation scheme for class Steiner tree in the special case with the real world background and highway interconnection problem. Close the gap between the lower bound and the upper bound for the approximation performance ratio of bottleneck Steiner tree in the Euclidean plane and make clear whether there exists a polynomial-time approximation scheme for the Steiner tree with minimum number of Steiner points and bounded edge-length. Implement efficient approximation algorithms to meet the requests from industries.