When Addition Becomes Optimization

This didactic paper introduces tropical algebra through a concrete running example, showing how changing the underlying arithmetic changes the meaning of linear algebra. In the min-plus setting, matrix products and powers become shortest-path, relaxation, and dynamic-programming computations; in the max-plus setting, the same algebraic structure describes critical-path timing, earliest-start propagation, and schedule evaluation. The paper connects these ideas to classical algorithms such as Bellman–Ford, Dijkstra, Floyd–Warshall, and Karp's algorithm, while emphasizing a practical distinction that is especially important in scheduling and operations research: tropical algebra is very effective at propagating costs, times, and lags through a structure that has already been chosen, and at locating its bottlenecks, but it does not by itself solve the harder combinatorial problem of selecting routes, assignments, or sequences. In this sense, the algebra provides a compact language for evaluating the consequences of fixed decisions, while the search over possible decisions remains a hard optimization problem.

The later sections extend this viewpoint from acyclic shortest-path and critical-path examples to schedule evaluation and to cyclic event systems. In cyclic max-plus dynamics, repeated propagation exposes long-run cadence behavior, max-plus eigenvalues, and bottleneck sensitivity, linking schedule timing to spectral structure. A closing benchmark study on a single laptop GPU explores when dense or heavily batched semiring matrix operations become computationally useful in practice. The paper is accompanied by interactive browser-based explainers and benchmark artifacts intended to support hands-on learning.