Existence of fast tropical matrix multiplication algorithms

Determine whether there exists a subcubic-time algorithm for multiplying matrices over the max-plus or min-plus semirings (tropical matrix multiplication) analogous to Strassen’s algorithm for matrices over fields, and, if so, construct such an algorithm and characterize its asymptotic complexity.

Background

Matrix multiplication over fields admits subcubic algorithms such as Strassen’s method, which significantly improves on the cubic bound. In tropical algebra (max-plus and min-plus semirings), many fundamental problems in optimization and graph theory are expressed via matrix operations, making efficient multiplication highly valuable.

The paper explicitly notes that, unlike the field case, no fast (Strassen-like) algorithm is known for tropical matrix multiplication, attributing difficulties to the noninvertibility of the additive operation. Establishing whether such an algorithm exists would have broad implications for tropical linear algebra and related applications.