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On the computational complexity of finding a sparse Wasserstein barycenter

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  • Steffen Borgwardt

    (University of Colorado Denver)

  • Stephan Patterson

    (Louisiana State University Shreveport)

Abstract

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of probability measures with finite support. In this paper, we show that finding a barycenter of sparse support is hard, even in dimension 2 and for only 3 measures. We prove this claim by showing that a special case of an intimately related decision problem SCMP—does there exist a measure with a non-mass-splitting transport cost and support size below prescribed bounds? Is NP-hard for all rational data. Our proof is based on a reduction from planar 3-dimensional matching and follows a strategy laid out by Spieksma and Woeginger (Eur J Oper Res 91:611–618, 1996) for a reduction to planar, minimum circumference 3-dimensional matching. While we closely mirror the actual steps of their proof, the arguments themselves differ fundamentally due to the complex nature of the discrete barycenter problem. Containment of SCMP in NP will remain open. We prove that, for a given measure, sparsity and cost of an optimal transport to a set of measures can be verified in polynomial time in the size of a bit encoding of the measure. However, the encoding size of a barycenter may be exponential in the encoding size of the underlying measures.

Suggested Citation

  • Steffen Borgwardt & Stephan Patterson, 2021. "On the computational complexity of finding a sparse Wasserstein barycenter," Journal of Combinatorial Optimization, Springer, vol. 41(3), pages 736-761, April.
    • Handle: RePEc:spr:jcomop:v:41:y:2021:i:3:d:10.1007_s10878-021-00713-5
    DOI: 10.1007/s10878-021-00713-5
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    References listed on IDEAS

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    1. Ethan Anderes & Steffen Borgwardt & Jacob Miller, 2016. "Discrete Wasserstein barycenters: optimal transport for discrete data," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(2), pages 389-409, October.
    2. Éva Tardos, 1986. "A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs," Operations Research, INFORMS, vol. 34(2), pages 250-256, April.
    3. Spieksma, Frits C. R. & Woeginger, Gerhard J., 1996. "Geometric three-dimensional assignment problems," European Journal of Operational Research, Elsevier, vol. 91(3), pages 611-618, June.
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