Conjectured lower bound for weighted APSP

Establish whether every algorithm for all-pairs shortest paths in weighted graphs requires, up to substantial improvements, the $(nm)$ running time obtained by running Dijkstra’s algorithm from all vertices.

Background

The paper invokes a conjectured barrier for weighted APSP to motivate the significance of decremental distance sensitivity oracles whose preprocessing matches the best known static APSP runtime. Confirming or refuting this conjectured lower bound would determine whether the oracle’s preprocessing complexity is essentially optimal.

References

In weighted graphs, it is conjectured that no algorithm for APSP exists which is substantially faster than the $(nm)$ time required for running Dijkstra's algorithm from all nodes.

— The Cost of Changing Edges for Diameter Computation and More  (2608.12628 - Hiken et al., 12 Aug 2026) in Section 1, “Some Wishful Thinking”