Mean-payoff Games and Parametric Tropical Two-sided Systems

We propose a general theory of parametric two-sided tropical linear systems based on the connections with zero-sum deterministic mean-payoff games. It is shown how this general theory specializes to particular tropical linear systems with parameter, arising from 1) the two-sided tropical eigenproblem Ax = λ+Bx and 2) the tropical linear programming. This involves 1) the problem of finding the spectrum of Ax = λ + Bx solved in pseudopolynomial time by the spectral function approach, and 2) bisection and Newton iteration methods for minimizing tropical linear functionals over tropical convex sets defined as solution sets of two-sided systems Ax ⊕ c ≤ Bx ⊕ d. Particular new results include a) explicit formula for the unique eigenvalue of symmetric two-sided igenproblem, b) generalized bound for bisection method for the fractional tropical linear programming, c) comparison between bisecton and Newton methods solving tropical linear programming problem.