Optimal Coordination Mechanisms for Unrelated Machine Scheduling

Yossi Azar, Lisa Fleischer, Kamal Jain, Vahab Mirrokni, Zoya Svitkina
2015 Operations Research  
We investigate the influence of different algorithmic choices on the approximation ratio in selfish scheduling. Our goal is to design local policies that minimize the inefficiency of resulting equilibria. In particular, we design optimal coordination mechanisms for unrelated machine scheduling, and improve the known approximation ratio from Θ(m) to Θ(log m), where m is the number of machines. A local policy for each machine orders the set of jobs assigned to it only based on parameters of those
more » ... jobs. A strongly local policy only uses the processing time of jobs on the the same machine. We prove that the approximation ratio of any set of strongly local ordering policies in equilibria is at least Ω(m). In particular, it implies that the approximation ratio of a greedy shortest-first algorithm for machine scheduling is at least Ω(m). This closes the gap between the known lower and upper bounds for this problem, and answers an open question raised by Ibarra and Kim [16] , and Davis and Jaffe [10]. We then design a local ordering policy with the approximation ratio of Θ(log m) in equilibria, and prove that this policy is optimal among all local ordering policies. This policy orders the jobs in the non-decreasing order of their inefficiency, i.e, the ratio between the processing time on that machine over the minimum processing time. Finally, we show that best responses of players for the inefficiency-based policy may not converge to a pure Nash equilibrium, and present a Θ(log 2 m) policy for which we can prove fast convergence of best responses to pure Nash equilibria. * azar@tau.ac.il. Microsoft Research, Redmond and Tel-Aviv University, Tel-Aviv, 69978, Israel. Research supported in part by the Israel Science Foundation. † kamalj@microsoft.com. Microsoft Research, Redmond. ‡ mirrokni@microsoft.com. Microsoft Research, Redmond. 1 For the definition of non-preemptive policies and the IIA property, see Section 2.
doi:10.1287/opre.2015.1363 fatcat:vso7rksv7ngwxh3vf43rbp5kte