Complexity of tropical and min-plus linear prevarieties

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 the vector $x$ is a solution to a polynomial $g1(x) \oplus g2(x) \oplus...\oplus gk(x)$, where $gi(x)$'s are tropical monomials, if the minimum in $\mini(g{i}(x))$ is attained at least twice. In min-plus algebra solutions of systems of equations of the form $g1(x)\oplus...\oplus gk(x) = h1(x)\oplus...\oplus hl(x)$ 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 reduction 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 are $\mathsf{NP}$-complete. We also extend some of our results to the systems of min-plus linear inequalities.