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Contention Resolution, Matrix Scaling and Fair Allocation

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Approximation and Online Algorithms (WAOA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12982))

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Abstract

A contention resolution (CR) scheme is a basic algorithmic primitive, that deals with how to allocate items among a random set S of competing players, while maintaining various properties. We consider the most basic setting where a single item must be allocated to some player in S. Here, in a seminal work, Feige and Vondrak (2006) designed a fair CR scheme when the set S is chosen from a product distribution. We explore whether such fair schemes exist for arbitrary non-product distributions on sets of players S, and whether they admit simple algorithmic primitives. Moreover, can we go beyond fair allocation and design such schemes for all possible achievable allocations.

We show that for any arbitrary distribution on sets of players S, and for any achievable allocation, there exist several natural CR schemes that can be succinctly described, are simple to implement and can be efficiently computed to any desired accuracy. We also characterize the space of achievable allocations for any distribution, give algorithms for computing an optimum fair allocation for arbitrary distributions, and describe other natural fair CR schemes for product distributions. These results are based on matrix scaling and various convex programming relaxations.

Part of the work has been done while the author was a postdoctoral fellow at CWI Amsterdam and TU Eindhoven, and was supported by the ERC Consolidator Grant 617951 and the NWO VICI grant 639.023.812.

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Notes

  1. 1.

    Define a distribution \(\tilde{\mathcal {P}}\) on \([n+1]\) players, where for \(S\subseteq [n+1]\), we set \(\tilde{p}_S = p_{S \setminus \{n+1\}} \) if \(n+1 \in S \) and \(\tilde{p}_S=0\) otherwise, and \(g_{n+1} = 1-p_\emptyset -\sum _{i \in [n]} g_i\). Then \(\sum _{i \in [n+1]} g_i=1-p_\emptyset \) and whenever the item is assigned to player \(n+1\) in the CR scheme for \(\tilde{\mathcal {P}}\) we do not assign it at all in the scheme for \(\mathcal {P}\).

  2. 2.

    Suppose \(n>3\) and \(S=\{1\}\) or \(S = \{1,\ldots ,n\}\), each with probability 1/2. Then \(q_1 = 1\) and \(q_i=1/2\) for \(i\ge 2\). For a balanced allocation with \(g=\alpha q\), the best g is \((1/(n-1),1/2(n-1),\ldots ,1/2(n-1))\), but here \(\sum _i g_i < 1-p_\emptyset \).

  3. 3.

    In the example above, this allows one to achieve the allocation \((1/2,1/2(n-1),\ldots ,1/2(n-1))\), which is clearly better.

  4. 4.

    It can always be computed efficiently to any desired accuracy \(\varepsilon >0\) using \(\text {poly}(n,1/\varepsilon )\) samples from \(\mathcal {P}\).

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Authors and Affiliations

  1. TU Eindhoven and CWI, Amsterdam, The Netherlands

    Nikhil Bansal

  2. Bar-Ilan University, Ramat Gan, Israel

    Ilan Reuven Cohen

Authors
  1. Nikhil Bansal
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  2. Ilan Reuven Cohen
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Correspondence to Nikhil Bansal or Ilan Reuven Cohen .

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Editors and Affiliations

  1. University of Waterloo, Waterloo, ON, Canada

    Jochen Koenemann

  2. RWTH Aachen University, Aachen, Germany

    Britta Peis

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Bansal, N., Cohen, I.R. (2021). Contention Resolution, Matrix Scaling and Fair Allocation. In: Koenemann, J., Peis, B. (eds) Approximation and Online Algorithms. WAOA 2021. Lecture Notes in Computer Science(), vol 12982. Springer, Cham. https://doi.org/10.1007/978-3-030-92702-8_16

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  • DOI: https://doi.org/10.1007/978-3-030-92702-8_16

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