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Inner product spaces for MinSum coordination mechanisms
Conference proceeding   Open access   Peer reviewed

Inner product spaces for MinSum coordination mechanisms

Richard Cole, José R. Correa, Vasilis Gkatzelis, Vahab Mirrokni, Neil Olver and ACM SIGACT
Proceedings of the forty-third annual ACM symposium on Theory of computing, pp 539-548
06 Jun 2011
DOI: https://doi.org/10.1145/1993636.1993708
Featured in Collection :   UN Sustainable Development Goals @ Drexel
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https://doi.org/10.1145/1993636.1993708View
Published, Version of Record (VoR) Open

Abstract

Theory of computation -- Design and analysis of algorithms -- Approximation algorithms analysis -- Scheduling algorithms Theory of computation -- Design and analysis of algorithms -- Online algorithms -- Online learning algorithms -- Scheduling algorithms Theory of computation -- Theory and algorithms for application domains -- Machine learning theory -- Reinforcement learning -- Sequential decision making
We study coordination mechanisms aiming to minimize the weighted sum of completion times of jobs in the context of selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obtain these approximation bounds, we introduce a new technique that while conceptually simple, seems to be quite powerful. The method entails mapping strategy vectors into a carefully chosen inner product space; costs are shown to correspond to the norm in this space, and the Nash condition also has a simple description. With this structure in place, we are able to prove a number of results, as follows. First, we consider Smith's Rule, which orders the jobs on a machine in ascending processing time to weight ratio, and show that it achieves an approximation ratio of 4. We also demonstrate that this is the best possible for deterministic non-preemptive strongly local policies. Since Smith's Rule is always optimal for a given fixed assignment, this may seem unsurprising, but we then show that better approximation ratios can be obtained if either preemption or randomization is allowed. We prove that ProportionalSharing, a preemptive strongly local policy, achieves an approximation ratio of 2.618 for the weighted sum of completion times, and an approximation ratio of 2.5 in the unweighted case. We also observe that these bounds are tight. Next, we consider Rand, a natural non-preemptive but randomized policy. We show that it achieves an approximation ratio of at most 2.13; moreover, if the sum of the weighted completion times is negligible compared to the cost of the optimal solution, this improves to π/2. Finally, we show that both ProportionalSharing and Rand induce potential games, and thus always have a pure Nash equilibrium (unlike Smith's Rule). This allows us to design the first combinatorial constant-factor approximation algorithm minimizing weighted completion time for unrelated machine scheduling. It achieves a factor of 2+ε for any ε > 0, and involves imitating best response dynamics using a variant of ProportionalSharing as the policy.

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28 citations in Web of Science
38 citations in Scopus

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#10 Reduced Inequalities

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Web of Science research areas
Computer Science, Theory & Methods
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