Abstract
We investigate coordination mechanisms that schedule n jobs on m unrelated machines. The objective is to minimize the latest completion of all jobs, i.e., the makespan. It is known that if the mechanism is non-preemptive, the price of anarchy is Ω(logm). Both Azar, Jain, and Mirrokni (SODA 2008) and Caragiannis (SODA 2009) raised the question whether it is possible to design a coordination mechanism that has constant price of anarchy using preemption. We give a negative answer.
All deterministic coordination mechanisms, if they are symmetric and satisfy the property of independence of irrelevant alternatives, even with preemption, have the price of anarchy \(\Omega(\frac{\log m}{\log \log m})\). Moreover, all randomized coordination mechanisms, if they are symmetric and unbiased, even with preemption, have similarly the price of anarchy \(\Omega(\frac{\log m}{\log \log m})\).
Our lower bound complements the result of Caragiannis, whose bcoord mechanism guarantees \(O(\frac{\log m}{\log \log m})\) price of anarchy. Our lower bound construction is surprisingly simple. En route we prove a Ramsey-type graph theorem, which can be of independent interest.
On the positive side, we observe that our lower bound construction critically uses the fact that the inefficiency of a job on a machine can be unbounded. If, on the other hand, the inefficiency is not unbounded, we demonstrate that it is possible to break the \(\Omega(\frac{\log m}{\log \log m})\) barrier on the price of anarchy by using known coordination mechanisms.
Part of this work is based on the first author’s Master thesis in Max-Planck-Institut für Informatik.
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Abed, F., Huang, CC. (2012). Preemptive Coordination Mechanisms for Unrelated Machines. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_3
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DOI: https://doi.org/10.1007/978-3-642-33090-2_3
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