The Price of Anarchy for Minsum Related Machine Scheduling
We address the classical uniformly related machine scheduling problem with minsum objective. The problem is solvable in polynomial time by the algorithm of Horowitz and Sahni. In that solution, each machine sequences its jobs shortest first. However when jobs may choose the machine on which they are processed, while keeping the same sequencing rule per machine, the resulting Nash equilibria are in general not optimal. The price of anarchy measures this optimality gap. By means of a new characterization of the optimal solution, we show that the price of anarchy in this setting is bounded from above by 2. We also give a lower bound of e/(e − 1) ≈ 1.58. This complements recent results on the price of anarchy for the more general unrelated machine scheduling problem, where the price of anarchy equals 4. Interestingly, as Nash equilibria coincide with shortest processing time first (SPT) schedules, the same bounds hold for SPT schedules. Thereby, our work also fills a gap in the literature.
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- 2.Azar, Y., Jain, K., Mirrokni, V.: (Almost) optimal coordination mechanisms for unrelated machine scheduling. In: Proceedings 19th SODA, pp. 323–332. ACM/SIAM (2008)Google Scholar
- 4.Cole, R., Correa, J.R., Gkatzelis, V., Mirrokni, V., Olver, N.: Inner Product Spaces for MinSum Coordination Mechanisms. In: Proceedings 43rd STOC, pp. 539–548. ACM (2011)Google Scholar
- 6.Correa, J., Queyranne, M.: Efficiency of Equilibria in Restricted Uniform Machine Scheduling with MINSUM Social Cost (manuscript) (2010)Google Scholar
- 7.Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Trans. Algorithms 3(1), Art. 4, 17 (2007)Google Scholar
- 16.Papadimitriou, C.: Algorithms, games, and the internet. In: Proceedings 33rd STOC, pp. 749–753. ACM (2001)Google Scholar
- 17.Roughgarden, T.: Intrinsic robustness of the price of anarchy. In: Proceedings 41st STOC, pp. 513–522. ACM (2009)Google Scholar