Complexity of comparing expressions in max-plus algebra

Abstract

Max-plus algebra has been widely used in the study of discrete-event dynamic systems. Using max-plus algebra makes it easy to specify safety constraints on events since they can be described as a set of inequalities of state variables, i.e., firing times of relevant events. This paper proves that the problem of solving max-plus inequalities in a cube (MAXINEQ) is nondeterministic polynomial-time hard (NP-hard) in strong sense and the problem of verifying max-plus inequalities (VERMAX-INEQ) is co-NP. As a corollary, the problem of solving a system of multivariate max-algebraic polynomial equalities and inequalities (MPEI) is shown to be NP-hard in strong sense. The results indicate the difficulties in comparing max-plus formulas in general. Problem structures of specific systems have to be explored to enable the development of efficient algorithms.