Complexity of Tropical and Min-plus Linear Prevarieties

Abstract

A tropical (or min-plus) semiring is a set Z (or Z ∪ {∞}) endowed with two operations: ⊕, which is just usual minimum, and ⊙, which is usual addition. In tropical algebra the vector x is a solution to a polynomial, where 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 formIn this paper we consider computational problems related to tropical linear system. We show that the solvability problem (both over Z and Z∪{∞}) and the problem of deciding the equivalence of two linear systems (both over Z and Z ∪ {∞}) are equivalent under polynomialtime reduction to mean payoff games and are also equivalent to analogous problems in min-plus algebra. In particular, all these problems belong to NP ∩ 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 NP-complete.We also extend some of our results to the systems of min-plus linear inequalities.

Approximating the volume of tropical polytopes is difficult

“…P3 reduces to P2: This follows from Corollary 8 of [GP15], using again the equivalence in Theorem 2.7.…”

Public key cryptography with max-plus matrices and polynomials

“…It is known (see [4]) that two-sided systems are polynomially equivalent to mean payoff games, which is a hard problem in NP CO-NP. For more details we refer to: [2], [13] and [14]. Finally, what are the advantages of max-plus semirings?…”

Tropical Newton–Puiseux Polynomials

“…Each of these polyhedra (including faces of all the dimensions) is defined by specifying the monomials of f i , 1 ≤ i ≤ k (treated as linear functions) on which the minima are attained (cf. e. g. [4]). The algorithm can find the partition of R n into polyhedra defined by given feasible tuples of signs (i. e. either the positive, either the negative or zero) of all the differences of the monomials of f i , 1 ≤ i ≤ k (in other words, by all the feasible orderings of the monomials of f i , 1 ≤ i ≤ k).…”

“…Observe that the validity of (5) would not change if one multiplies all the rational coefficients in T rop(F i ), 0 ≤ i ≤ d by their common denominator m and simultaneously all T rop(A I ) (see(4)) by m to make all the coefficients in T rop(F i ), 0 ≤ i ≤ d integers.…”