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Lower Bounds for Tropical Circuits and Dynamic Programs

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Abstract

Tropical circuits are circuits with Min and Plus, or Max and Plus operations as gates. Their importance stems from their intimate relation to dynamic programming algorithms. The power of tropical circuits lies somewhere between that of monotone boolean circuits and monotone arithmetic circuits. In this paper we present some lower bounds arguments for tropical circuits, and hence, for dynamic programs.

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Notes

  1. 1 There is nothing special about the term “tropical”. Simply, this term is used in honor of Imre Simon who lived in Sao Paulo (south tropic). Tropical algebra and tropical geometry are now intensively studied topics in mathematics.

  2. 2 Usually, polynomials of more than one variable are called multivariate, but we will omit this for shortness.

  3. 3 We will always denote circuits as upright letters F,G,H,…, and their produced polynomials by italic versions F,G,H,….

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Acknowledgments

I am thankful to Dima Grigoriev, Georg Schnitger and Igor Sergeev for interesting discussions.

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Authors and Affiliations

  1. Institute of Computer Science, University of Frankfurt, Frankfurt am Main, Germany

    Stasys Jukna

  2. Institute of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania

    Stasys Jukna

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  1. Stasys Jukna
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Correspondence to Stasys Jukna.

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Research supported by the DFG grant SCHN 503/6-1.

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Jukna, S. Lower Bounds for Tropical Circuits and Dynamic Programs. Theory Comput Syst 57, 160–194 (2015). https://doi.org/10.1007/s00224-014-9574-4

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