Enforce shortest-chain constraints in the min-cost-flow reduction

Develop a min-cost-flow reduction that efficiently constructs chain covers whose chains are shortest paths between their endpoints, thereby enabling faster construction of directed (1+ε)-hopsets with improved size–hopbound tradeoffs.

Background

The faster shortcut-set algorithm of Kogan and Parter uses a min-cost-flow reduction to construct chain covers without explicitly computing the transitive closure. For approximate hopsets, however, the chains must additionally correspond to shortest paths between their endpoints so that the resulting hopset preserves distances within the required approximation factor.

The paper identifies the inability to impose this shortest-chain condition within the existing min-cost-flow framework as the central obstacle to obtaining faster algorithms for the improved directed hopset size–hopbound tradeoffs. The paper resolves the algorithmic bottleneck by using a different greedy construction, so this particular min-cost-flow question remains unresolved.

References

The main challenge is that for hopsets we need these chain covers to also correspond to shortest paths between their endpoints, and we do not know how to enforce such a requirement in the min cost flow reduction of .

— Faster Directed Hopsets, Distance Preservers, and Deterministic Shortcut Sets  (2610.12057 - Bals et al., 8 Oct 2026) in Section 1, Introduction, subparagraph “Faster Hopsets”