Two Lectures on Max-plus Algebra

Exotic semirings such as the “(max,+) semiring” (R ∪ {−∞},max,+), or the “tropical semiring” (N ∪ {+∞},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities). Despite this apparent profusion, there is a small set of common, non-naive, basic results and problems, in general not known outside the (max,+) community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core of (max,+) results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of discrete event systems, analysis of Markov decision processes with average cost). Basic techniques include: solving all kinds of systems of linear equations, sometimes with exotic symmetrization and determinant techniques; using the (max,+) Perron-Frobenius theory to study the dynamics of (max,+) linear maps. We point out some open problems and current developments.