Whether Inner Product captures natural dynamic graph problems

Determine whether the multiphase problem with Inner Product captures natural dynamic graph problems such as dynamic reachability, shortest paths, or connectivity, thereby permitting the corresponding lower-bound framework to apply directly to those problems.

Background

The paper contrasts two versions of the multiphase problem: one based on Inner Product, for which a recent framework achieves an \Omega((\log n/\log\log n)2) lower bound, and one based on Disjointness, which is connected by known reductions to natural dynamic graph problems.

The authors explicitly note that it is not known whether the Inner Product formulation captures natural dynamic graph problems. The present work therefore transfers the framework from Inner Product to Disjointness rather than resolving the applicability of Inner Product itself.

References

Inner Product, however, is not known to capture natural dynamic graph problems: a's reductions to reachability and related problems start from Disjointness.

— $Ω((\log n/\log\log n)^2)$ Lower Bounds for Dynamic Graph Problems  (2610.10904 - Ko, 7 Oct 2026) in Section 1, subsection “A new landscape,” final paragraph