computational complexity

, Volume 24, Issue 1, pp 31–64 | Cite as

Complexity of Tropical and Min-plus Linear Prevarieties

  • Dima Grigoriev
  • Vladimir V. Podolskii


A tropical (or min-plus) semiring is a set \({\mathbb{Z}}\) (or \({\mathbb{Z \cup \{\infty\}}}\)) endowed with two operations: \({\oplus}\) , which is just usual minimum, and \({\odot}\) , which is usual addition. In tropical algebra, a vector x is a solution to a polynomial \({g_1(x) \oplus g_2(x) \oplus \cdots \oplus g_k(x)}\) , where the g i (x)s are tropical monomials, if the minimum in min i (g i (x)) is attained at least twice. In min-plus algebra solutions of systems of equations of the form \({g_1(x)\oplus \cdots \oplus g_k(x) = h_1(x)\oplus \cdots \oplus h_l(x)}\) are studied.

In this paper, we consider computational problems related to tropical linear system. We show that the solvability problem (both over \({\mathbb{Z}}\) and \({\mathbb{Z} \cup \{\infty\}}\)) and the problem of deciding the equivalence of two linear systems (both over \({\mathbb{Z}}\) and \({\mathbb{Z} \cup \{\infty\}}\)) are equivalent under polynomial-time reductions to mean payoff games and are also equivalent to analogous problems in min-plus algebra. In particular, all these problems belong to \({\mathsf{NP}\cap \mathsf{coNP}}\) . Thus, we provide a tight connection of computational aspects of tropical linear algebra with mean payoff games and min-plus linear algebra. On the other hand, we show that computing the dimension of the solution space of a tropical linear system and of a min-plus linear system is \({\mathsf{NP}}\) -complete.


Tropical linear systems min-plus linear systems tropical prevarieties min-plus prevarieties mean payoff games 

Subject classification

15A80 68Q17 91A43 


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.CNRS, MathématiquesUniversité de LilleVilleneuve d’AscqFrance
  2. 2.Steklov Mathematical InstituteMoscowRussia

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