IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v41y2021i3d10.1007_s10878-021-00713-5.html
   My bibliography  Save this article

On the computational complexity of finding a sparse Wasserstein barycenter

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-021-00713-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10878-021-00713-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Steffen Borgwardt, 2022. "An LP-based, strongly-polynomial 2-approximation algorithm for sparse Wasserstein barycenters," Operational Research, Springer, vol. 22(2), pages 1511-1551, April.
    2. Ting Pong & Hao Sun & Ningchuan Wang & Henry Wolkowicz, 2016. "Eigenvalue, quadratic programming, and semidefinite programming relaxations for a cut minimization problem," Computational Optimization and Applications, Springer, vol. 63(2), pages 333-364, March.
    3. R. B. Bapat & S. K. Neogy, 2016. "On a quadratic programming problem involving distances in trees," Annals of Operations Research, Springer, vol. 243(1), pages 365-373, August.
    4. Huang, Edward & Mital, Pratik & Goetschalckx, Marc & Wu, Kan, 2016. "Optimal assignment of airport baggage unloading zones to outgoing flights," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 94(C), pages 110-122.
    5. Amitai Armon & Iftah Gamzu & Danny Segev, 2014. "Mobile facility location: combinatorial filtering via weighted occupancy," Journal of Combinatorial Optimization, Springer, vol. 28(2), pages 358-375, August.
    6. Balaji Gopalakrishnan & Seunghyun Kong & Earl Barnes & Ellis Johnson & Joel Sokol, 2011. "A least-squares minimum-cost network flow algorithm," Annals of Operations Research, Springer, vol. 186(1), pages 119-140, June.
    7. Clemens Heuberger, 2004. "Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 329-361, September.
    8. Amitabh Basu & Jesús A. De Loera & Mark Junod, 2014. "On Chubanov's Method for Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 26(2), pages 336-350, May.
    9. László A. Végh, 2017. "A Strongly Polynomial Algorithm for Generalized Flow Maximization," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 179-211, January.
    10. Mao-Cheng Cai & Xiaoguang Yang & Yanjun Li, 1999. "Inverse Polymatroidal Flow Problem," Journal of Combinatorial Optimization, Springer, vol. 3(1), pages 115-126, July.
    11. Ilan Adler & Martin Bullinger & Vijay V. Vazirani, 2024. "A Generalization of von Neumann's Reduction from the Assignment Problem to Zero-Sum Games," Papers 2410.10767, arXiv.org.
    12. D. V. Gribanov & D. S. Malyshev & P. M. Pardalos & S. I. Veselov, 2018. "FPT-algorithms for some problems related to integer programming," Journal of Combinatorial Optimization, Springer, vol. 35(4), pages 1128-1146, May.
    13. M. Cai & X. Yang & Y. Li, 2000. "Inverse Problems of Submodular Functions on Digraphs," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 559-575, March.
    14. Prabhjot Kaur & Anuj Sharma & Vanita Verma & Kalpana Dahiya, 2022. "An alternate approach to solve two-level hierarchical time minimization transportation problem," 4OR, Springer, vol. 20(1), pages 23-61, March.
    15. Paulo Oliveira, 2014. "A strongly polynomial-time algorithm for the strict homogeneous linear-inequality feasibility problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(3), pages 267-284, December.
    16. Orlin, James B., 1953-., 1988. "A faster strongly polynomial minimum cost flow algorithm," Working papers 2042-88., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    17. Cai Mao-Cheng, 1999. "Inverse Problems of Matroid Intersection," Journal of Combinatorial Optimization, Springer, vol. 3(4), pages 465-474, December.
    18. Puerto, Justo & Tamir, Arie & Perea, Federico, 2011. "A cooperative location game based on the 1-center location problem," European Journal of Operational Research, Elsevier, vol. 214(2), pages 317-330, October.
    19. H-J Bandelt & A Maas & F C R Spieksma, 2004. "Local search heuristics for multi-index assignment problems with decomposable costs," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(7), pages 694-704, July.
    20. Carsten Gottschlich & Dominic Schuhmacher, 2014. "The Shortlist Method for Fast Computation of the Earth Mover's Distance and Finding Optimal Solutions to Transportation Problems," PLOS ONE, Public Library of Science, vol. 9(10), pages 1-10, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:41:y:2021:i:3:d:10.1007_s10878-021-00713-5. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.