Signed Tropical Convexity

Authors Georg Loho, László A. Végh

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Georg Loho
  • London School of Economics and Political Science, UK
László A. Végh
  • London School of Economics and Political Science, UK


We thank Daniel Dadush for insisting on balanced numbers and for inspiring discussions on the tropical Fourier-Motzkin elimination. We thank Xavier Allamigeon as well as Stéphane Gaubert for communicating their related work. We are grateful to Matthias Schymura for helping to get an intuition for the unintuitive line segments. Furthermore, we thank Mateusz Skomra and an anonymous referee for pointing out the NP-completeness stated in Theorem 4.21.

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Georg Loho and László A. Végh. Signed Tropical Convexity. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 24:1-24:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We establish a new notion of tropical convexity for signed tropical numbers. We provide several equivalent descriptions involving balance relations and intersections of open halfspaces as well as the image of a union of polytopes over Puiseux series and hyperoperations. Along the way, we deduce a new Farkas' lemma and Fourier-Motzkin elimination without the non-negativity restriction on the variables. This leads to a Minkowski-Weyl theorem for polytopes over the signed tropical numbers.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • tropical convexity
  • signed tropical numbers
  • Farkas' lemma


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