An Improved Randomized Truthful Mechanism for Scheduling Unrelated Machines

arXiv:0802.2851v1 [cs.DS] 20 Feb 2008 Symposium on Theoretical Aspects of Computer Science 2008 (Bordeaux), pp. 527-538 www.stacs-conf.org AN IMPROVED RANDOMIZED TRUTHFUL MECHANISM FOR SCHEDULING UNRELATED MACHINES PINYAN LU 1 AND CHANGYUAN YU 1 1 Institute for Theoretical Computer Science, Tsinghua University, Beijing, 100084, P.R. China E-mail address: {lpy,yucy05}@mails.tsinghua.edu.cn URL: http://www.itcs.tsinghua.edu.cn/{LuPY,YuCY} Abstract. We study the scheduling problem on unrelated machines in the mechanism design setting. This problem was proposed and studied in the seminal paper of Nisan and Ronen [NR99], where they gave a 1.75-approximation randomized truthful mechanism for the case of two machines. We improve this result by a 1.6737-approximation randomized truthful mechanism. We also generalize our result to a 0.8368m-approximation mechanism for task scheduling with m machines, which improve the previous best upper bound of 0.875m[MS07]. 1. Introduction Mechanism design has become an active area of research both in Computer Science and Game Theory. In the mechanism design setting, players are selfish and wish to maximize their own utilities. To deal with the selfishness of the players, a mechanism should both satisfy some game-theoretical requirements such as truthfulness and some computational properties such as good approximation ratio. The study of their algorithmic aspect was ini- tiated by Nisan and Ronen in their seminal paper “Algorithmic Mechanism Design” [NR99]. The focus of that paper was on the scheduling problem on unrelated machines, for which the standard mechanism design tools ( VCG mechanisms [Clarke71, Groves1973, Vickrey61])do not suffice. They proved that no deterministic mechanism can have an approximation ratio better than 2 for this problem. This bound is tight for the case of two machines. How- ever if we allow randomized mechanisms, this bound can be beaten. In particular they gave a 1.75-approximation randomized truthful mechanism for the case of two machines. Since then, many researchers have studied the scheduling problem on unrelated machines in mechanism design setting [JP99, Sourd01, SS02, SX02, GMW07, CKV07, CKK07, MS07]. However their mechanism remains the best to the best of our knowledge. In a recent paper [MS07], Mu’alem and Schapira proved a lower bound of 1.5 for this setting. So to explore the exact bound between 1.5 and 1.75 is an interesting open problem in this area. In this paper, 1998 ACM Subject Classification: algorithmic mechanism design. Key words and phrases: truthful mechanism, scheduling. Supported by the National Natural Science Foundation of China Grant 60553001 and the National Basic Research Program of China Grant 2007CB807900, 2007CB807901. c ⃝ P. Lu and C. Yu CC ⃝ Creative Commons Attribution-NoDerivs License 528 P. LU AND C. YU we improve the upper bound from 1.75 to 1.6737. Formally we give a 1.6737-approximation randomized truthful mechanism for task scheduling with two machines. Using similar tech- niques of [MS07], we also generalize our result to a 0.8368m-approximation mechanism for task scheduling with m machines. Let us describe the problem more carefully. There are m machines and n tasks, and each machine is controlled by an agent. We use ti j to denote the running time of task j on machine i, which is also called the type value of the agent(machine) i on task j. The objective is to minimize the completion time of the last assignment (the makespan). Unlike in the classical optimization problem, the scheduling designer does not know ti j. Each selfish agent i holds his/her own type values (the ti js). In order to motivate the agents to report their true value ti js, the mechanism needs to pay the agents. So a mechanism consists of an allocation algorithm and a payment algorithm. A mechanism is called truthful when telling one’s true value is among the optimal strategies for each agent, no matter how other agents behave. Here the utility of each agent is the payment he/she gets minus the load of tasks allocated to his/her machine. When randomness is involved, there are two versions of truthfulness: in the stronger version, i.e. universally truthfulness, the mechanism remains truthful even if the agents know the random bits; in the weaker version, i.e. truthfulness in expectation, an agent maximizes his/her expected utility by telling the true type value. Our mechanisms proposed in this paper are universally truthful. Now we can talk about the high level idea of the technical part. Here we only talk about the allocation algorithms, and the corresponding payment algorithms, which make the mechanism truthful, will be given later. First we describe Nisan and Ronen’s mechanism [NR99]. In their mechanism, each task is allocated independently. For a particular task j, if the two values t1 j and t2 j are relatively close to each other, say t1 j/t2 j ∈[3/4, 4/3], then they allocate task j randomly to machine 1 or 2 with equal probability; if o

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