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The Online Best Reply Algorithm for Resource Allocation Problems

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Algorithmic Game Theory (SAGT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 11801))

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Abstract

We study the performance of a best reply algorithm for online resource allocation problems with a diseconomy of scale. In an online resource allocation problem, we are given a set of resources and a set of requests that arrive in an online manner. Each request consists of a set of feasible allocations and an allocation is a set of resources. The total cost of an allocation vector is given by the sum of the resources’ costs, where each resource’s cost depends on the total load on the resource under the allocation vector. We analyze the natural online procedure where each request is allocated greedily to a feasible set of resources that minimizes the individual cost of that particular request. In the literature, this algorithm is also known as a one-round walk in congestion games starting from the empty state. For unweighted resource allocation problems with polynomial cost functions with maximum degree d, upper bounds on the competitive ratio of this greedy algorithm were known only for the special cases \(d\in \{1, 2, 3\}\). In this paper, we show a general upper bound on the competitive ratio of \(d(d / W(\frac{1.2d-1}{d+1}))^{d+1}\) for the unweighted case where W denotes the Lambert-W function on \(\mathbb {R}_{\ge 0}\). For the weighted case, we show that the competitive ratio of the greedy algorithm is bounded from above by \((d/W(\frac{d}{d+1}))^{d+1}\).

A. Tönnis—Partially supported by CONICYT grant PCI PII 20150140 and ERC Starting Grant 306465 (BeyondWorstCase).

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Author information

Authors and Affiliations

  1. Operations Research, HU Berlin, Berlin, Germany

    Max Klimm

  2. Institute for Computer Science, Goethe University Frankfurt, Frankfurt, Germany

    Daniel Schmand

  3. Department of Computer Science, University of Bonn, Bonn, Germany

    Andreas Tönnis

Authors
  1. Max Klimm
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  2. Daniel Schmand
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  3. Andreas Tönnis
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Corresponding author

Correspondence to Daniel Schmand .

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Editors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    Dimitris Fotakis

  2. Athens University of Economics and Business, Athens, Greece

    Evangelos Markakis

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Klimm, M., Schmand, D., Tönnis, A. (2019). The Online Best Reply Algorithm for Resource Allocation Problems. In: Fotakis, D., Markakis, E. (eds) Algorithmic Game Theory. SAGT 2019. Lecture Notes in Computer Science(), vol 11801. Springer, Cham. https://doi.org/10.1007/978-3-030-30473-7_14

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  • DOI: https://doi.org/10.1007/978-3-030-30473-7_14

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-30473-7

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