$Ω((\log n/\log\log n)^2)$ Lower Bounds for Dynamic Graph Problems

2026-10-07

AI Summary

This study addresses dynamic graph problems such as DAG reachability, establishing an unconditional squared-logarithmic lower bound on the trade-off between query and update times. Methodologically, it extends Ko’s framework to the multi-stage Disjointness problem and replaces traditional discrepancy-based verification with a one-sided corruption bound, integrating communication complexity theory with combinatorial optimization techniques to complete the proof. This work achieves the strongest known time lower bounds for dynamic problems to date, significantly improving upon prior results by Larsen and Yu. By doing so, it provides new theoretical foundations for understanding the complexity limits of dynamic graph algorithms.

Abstract

We prove an $Ω((\log n/\log\log n)^2)$ unconditional lower bound on the maximum of the query time and update time for dynamic data structures supporting reachability queries in $n$-node directed acyclic graphs under edge insertions. This improves the $\widetildeΩ(\log^{3/2} n)$ lower bound of Larsen and Yu [SICOMP 2025], and matches the strongest lower bound known for any dynamic problem. The same bound holds for incremental undirected shortest paths and subgraph connectivity. To prove it, we bring the recent framework of Ko [FOCS 2026], which gave this bound for Pătraşcu's multiphase problem with Inner Product, to the multiphase problem with Disjointness. Our main technical contribution is to show that verification in Ko's 2.5-round multiphase communication game makes a one-sided corruption bound suffice in place of discrepancy, which Disjointness lacks.

Problem

• dynamic graph problems
• lower bounds
• reachability queries
• directed acyclic graphs
• shortest paths

Innovation

• dynamic graph problems
• unconditional lower bounds
• multiphase communication game
• one-sided corruption bound
• reachability queries