Complexity of constructing 2-shortcuts

Determine the computational complexity of constructing a 2-shortcut for a directed graph, thereby resolving whether 2-shortcuts can be computed asymptotically faster than transitive closure.

Background

A d-shortcut is a subset of the transitive closure whose addition reduces the graph diameter to at most d. Constructing a 1-shortcut is computationally equivalent to computing the entire transitive closure, while the paper develops faster constructions for d=3 and for even d from 4 through O(log n).

The authors explicitly identify the case d=2 as unresolved, marking it as the first gap in their computational hierarchy of shortcut constructions. Resolving this problem would clarify how rapidly the construction complexity decreases as the permitted shortcut diameter increases.

References

While computing a $1$-shortcut is equivalent to computing the transitive closure, the complexity of computing a $2$-shortcut remains open.

— $\tilde{O}(1)$-Depth Parallel Reachability Faster than Transitive Closure  (2608.13231 - Kogan et al., 13 Aug 2026) in Section 1, subsection “A Hierarchy of Small-d Shortcut Algorithms”