November 27 – December 2 , 2016, Dagstuhl Seminar 16482

Algorithms and Effectivity in Tropical Mathematics and Beyond


Stéphane Gaubert (INRIA Saclay – Île-de-France, FR)
Dimitry Grigoriev (Lille I University, FR)
Michael Joswig (TU Berlin, DE)
Thorsten Theobald (Goethe-Universität Frankfurt am Main, DE)

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Dagstuhl Report, Volume 6, Issue 11 Dagstuhl Report
Aims & Scope
List of Participants
Dagstuhl's Impact: Documents available


Tropical mathematics is a uniting name for different research directions which involve the semi-ring of real numbers endowed with the operations min, + (called the tropical semi-ring). It has emerged in several areas of computer science and of pure and applied mathematics. For the first time, this seminar brought together the computer science and the mathematics viewpoints. A focus was on effective methods, algorithms and complexity bounds in tropical mathematics, and on their relations with open questions in various areas of computer science, including optimization, game theory and circuit complexity.

One of the oldest open algorithmic challenges in tropical mathematics is the complexity of solving systems of tropical linear equalities and inequalities. It is known to be equivalent to solving mean payoff games. The solvability of these problems is among the few known problems which are contained in the intersection NP cap co-NP, but not currently known to be in P. This leads to new approaches in linear programming or convex semialgebraic programming over nonarchimedean fields.

According to the organizers' points of view the seminar was quite successful. In addition to 28 talks there were many informal discussions and exchange of ideas in small groups. We expect several new common papers of the participants conceived during the seminar. An important feature was to bring together experts with different backgrounds who often knew other participants just by their publications. According to the opinions expressed, the participants learned a lot of new things. The seminar was especially useful for the young people.

Every talk, in addition to new results, also contained open problems. This created a lot of interaction in subsequent discussions. The audience was very active, many questions were posed to the speakers during the talks and the breaks.

  • Algorithmical problems of foundations of tropical mathematics (H. Markwig, D. Maclagan, F. Rincon, V. Podolskii);
  • Complexity of games and of tropical linear and convex algebra (M. Bodirsky, S. Gaubert, T. Hansen, M. Joswig, G. Loho, M. MacCaig, B. Schröter, S. Sergeev, M. Skomra);
  • Algorithms and complexity bounds on tropical varieties (F.Bihan, D. Grigoriev, S. Hampe, A. Jensen, T. Jörgens, L. Tabera, T. Theobald, T. de Wolff). We mention that S. Hampe has made a demonstration of the software Polymake for computations in tropical algebra;
  • Algorithms in tropical differential algebra (F. Aroca, C. Garay);
  • Interactions of tropical mathematics with algorithmic issues in classical mathematics (M. Akian, X. Allamigeon, Bo Lin, M. Rojas, C. Vinzant).

During the seminar a manuscript appeared (on the Internet) with the very strong claim of a quasi-polynomial complexity algorithm for parity games. It was a lucky coincidence that so many experts with various backgrounds were present. So a special evening session on Thursday was created to analyze this result and its ramifications. This was one more highlight.

Summary text license
  Creative Commons BY 3.0 Unported license
  Stéphane Gaubert, Dimitry Grigoriev, Michael Joswig, and Thorsten Theobald


  • Data Structures / Algorithms / Complexity
  • Optimization / Scheduling


  • Algorithms in tropical mathematics
  • Complexity
  • Effective bounds
  • Optimization
  • Zero-sum games


In the series Dagstuhl Reports each Dagstuhl Seminar and Dagstuhl Perspectives Workshop is documented. The seminar organizers, in cooperation with the collector, prepare a report that includes contributions from the participants' talks together with a summary of the seminar.


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Dagstuhl's Impact

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Furthermore, a comprehensive peer-reviewed collection of research papers can be published in the series Dagstuhl Follow-Ups.