Exact complexity of the maximum weight path problem

Determine the exact computational complexity of the maximum weight path problem with binary-encoded edge weights; specifically, given a directed graph with integer weights on edges and designated vertices s and t, establish the precise complexity class of computing the maximum-weight path from s to t.

Background

The authors’ algorithmic framework for directedness uses maximum-weight path computations via matrix powering over the max-plus semiring. While they obtain AC^1 procedures in their setting, they note that the general complexity of the maximum weight path problem (with binary-encoded weights) is unknown, referencing Cook’s taxonomy.

Clarifying the exact complexity of this path problem would sharpen the theoretical underpinnings of the paper’s AC^1 upper bounds and inform broader parallel complexity results for weighted path computations.