Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Combinatorial Algorithms

IWOCA 2023: Combinatorial Algorithms pp 111–122Cite as

A Polyhedral Perspective on Tropical Convolutions

  • 257 Accesses

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13889)

Abstract

Tropical (or min-plus) convolution is a well-studied algorithmic primitive in fine-grained complexity. We exhibit a novel connection between polyhedral formulations and tropical convolution, through which we arrive at a dual variant of tropical convolution. We show this dual operation to be equivalent to primal convolutions. This leads us to considering the geometric objects that arise from dual tropical convolution as a new approach to algorithms and lower bounds for tropical convolutions. In particular, we initiate the study of their extended formulations.

C. Brand was supported by the Austrian Science Fund (FWF, Project Y1329: ParAI). M. Koutecký was partially supported by Charles University project UNCE/SCI/004 and by the project 22-22997S of GA ČR. A. Lassota was supported by the Swiss National Science Foundation within the project Complexity of integer Programming (207365).

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   74.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Aprile, M., Fiorini, S., Huynh, T., Joret, G., Wood, D. R.: Smaller extended formulations for spanning tree polytopes in minor-closed classes and beyond. Electron. J. Comb. 28(4) (2021)

    Google Scholar 

  2. Bellman, R., Karush, W.: Mathematical programming and the maximum transform. J Soc. Ind. Appl. Math. 10(3), 550–567 (1962)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Bremner, D., et al.: Necklaces, convolutions, and X+Y. Algorithmica 69(2), 294–314 (2014)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Bussieck, M.R., Hassler, H., Woeginger, G.J., Zimmermann, U.T.: Fast algorithms for the maximum convolution problem. Oper. Res. Lett. 15(3), 133–141 (1994)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Chan, T.M., Lewenstein, M.: Clustered Integer 3SUM via Additive Combinatorics. In: Servedio, A.R., Rubinfeld, R. (eds.) ACM, pp. 31–40 (2015)

    Google Scholar 

  6. Cygan, M., Mucha, M., Wegrzycki, K., Wlodarczyk. , M.: On problems equivalent to (min, +)-convolution. ACM Trans. Algorithms 15(1), 14:1–14:25 (2019)

    Google Scholar 

  7. Fenchel, W., Blackett, W.D.: Convex cones, sets, and functions. Dept. Math. Logistics Res. Proj. (1953). Princeton University

    Google Scholar 

  8. Fiorini, S., Huynh, T., Joret, G., Pashkovich, K.: Smaller extended formulations for the spanning tree polytope of bounded-genus graphs. Discrete Comput. Geom. 57(3), 757–761 (2017). https://doi.org/10.1007/s00454-016-9852-9

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Fiorini, S., Massar, S., Pokutta, S., Tiwary, H. R., De Wolf, R.: Exponential Lower Bounds for Polytopes in Combinatorial Optimization. J. ACM 62(2), 17:1–17:23 (2015)

    Google Scholar 

  10. Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Künnemann, M., Paturi, R., Schneider, S.: On the fine-grained complexity of one-dimensional dynamic programming. In: Chatzigiannakis, i., Indyk, P., Kuhn, F., Muscholl, A. (eds.) 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, 10–14 July 2017, Warsaw, Poland, vol. 80. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 21:1–21:15 (2017)

    Google Scholar 

  12. Martin, R.K., Rardin, R.L., Campbell, B.A.: Polyhedral characterization of discrete dynamic programming. Oper. Res. 38(1), 127–138 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Rothvoss, T.: The matching polytope has exponential extension complexity. J. ACM 64(6), 41:1–41:19 (2017)

    Google Scholar 

  14. Williams, R.R.: Faster all-pairs shortest paths via circuit complexity. SIAM J. Comput. 47(5), 1965–1985 (2018)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Williams, R.: A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci. 348(2–3), 357–365 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Williams, V.V., Williams, R.R.: Subcubic equivalences between path, matrix, and triangle problems. J. ACM 65(5), 27:1–27:38 (2018)

    Google Scholar 

  17. Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Vienna University of Technology, Vienna, Austria

    Cornelius Brand

  2. Computer Science Institute, Charles University, Prague, Czech Republic

    Martin Koutecký

  3. Institute of Mathematics, EPFL, Lausanne, Switzerland

    Alexandra Lassota

Authors
  1. Cornelius Brand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Koutecký
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexandra Lassota
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cornelius Brand .

Editor information

Editors and Affiliations

  1. National Cheng Kung University, Tainan, Taiwan

    Sun-Yuan Hsieh

  2. National Taipei University of Business, Taoyuan, Taiwan

    Ling-Ju Hung

  3. National Taitung University, Taitung, Taiwan

    Chia-Wei Lee

Rights and permissions

Reprints and Permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Brand, C., Koutecký, M., Lassota, A. (2023). A Polyhedral Perspective on Tropical Convolutions. In: Hsieh, SY., Hung, LJ., Lee, CW. (eds) Combinatorial Algorithms. IWOCA 2023. Lecture Notes in Computer Science, vol 13889. Springer, Cham. https://doi.org/10.1007/978-3-031-34347-6_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-34347-6_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-34346-9

  • Online ISBN: 978-3-031-34347-6

  • eBook Packages: Computer ScienceComputer Science (R0)

search

Navigation