Directed hereditary edge discrepancy upper bound

Establish whether the hereditary edge discrepancy of the system of unique shortest paths in directed weighted graphs admits an O(n^{1/4}) upper bound, matching the O(n^{1/4}) bound known for hereditary vertex discrepancy and for undirected edge discrepancy via explicit colorings.

Background

The paper proves O(n^{1/4}) upper bounds for hereditary vertex discrepancy for unique shortest paths and provides explicit constructions achieving O(n^{1/4}) for edge discrepancy in undirected graphs and DAGs. For directed graphs, the existential bound for hereditary edge discrepancy obtained via shatter functions is only O(m^{1/4}), which can be loose for dense graphs.

They also prove lower bounds of order n^{1/4} (up to polylogarithmic factors) for hereditary discrepancy, showing the bound is tight for several settings. The remaining gap in the directed edge case motivates seeking an O(n^{1/4}) upper bound that depends on the number of vertices rather than edges.