Complexity of Tropical and Min-plus Linear Prevarieties

Abstract

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.