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A Polyhedral Perspective on Tropical Convolutions

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Combinatorial Algorithms (IWOCA 2023)

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

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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).

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Change history

  • 11 April 2024

    A correction has been published.

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), 4–47 (2021)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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)

    Article  MathSciNet  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

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

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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
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  2. Martin Koutecký
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  3. Alexandra Lassota
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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

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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

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

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-34346-9

  • Online ISBN: 978-3-031-34347-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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