Polynomial lower bound for the multiphase Disjointness problem

Establish that the multiphase problem with Disjointness requires polynomial time when the number of query sets satisfies m = n^{1+\Omega(1)}, and consequently establish polynomial unconditional lower bounds for dynamic reachability and the other dynamic problems obtained through the associated reductions.

Background

The paper studies the multiphase problem in which an initial family of sets is preprocessed, elements are inserted into a set X, and a query asks whether a designated set is disjoint from X. This problem was introduced as a source of conditional lower bounds for dynamic graph problems, including dynamic reachability.

The authors explain that a polynomial lower bound for the multiphase problem would yield polynomial unconditional cell-probe lower bounds for dynamic reachability and related dynamic graph problems. They state that the conjecture remains part of the motivation for studying stronger lower bounds, while the paper proves only an \Omega((\log n/\log\log n)2) lower bound.

References

Besides the conditional lower bounds above, a \ conjectured that the multiphase problem requires polynomial time, and identified a three-party communication game, now known as the multiphase communication game, whose hardness would imply this even in the cell-probe model, i.e. unconditionally; by his reductions, this would give polynomial unconditional lower bounds for dynamic reachability.

— $Ω((\log n/\log\log n)^2)$ Lower Bounds for Dynamic Graph Problems  (2610.10904 - Ko, 7 Oct 2026) in Section 1, subsection “Our results,” paragraph “The route: multiphase Disjointness” (also discussed in Section 1, subsection “Previous work,” paragraph “Unconditional lower bounds”)