Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs

TL;DR

This paper establishes the first conditional lower bounds for dynamic graph algorithms on specific graph families like constant-degree, power-law, and expander graphs, showing limitations on achieving fast updates and queries simultaneously.

Contribution

It provides novel lower bounds for dynamic algorithms on structured graph classes, extending known results from general graphs to practical, common graph families.

Findings

No dynamic algorithms with both sub-polynomial update and query times exist for these graph classes.

Lower bounds apply to problems like maximum matching and distance computation.

Results hold even for well-studied graph families such as expanders and power-law graphs.

Abstract

A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and ($s,t$)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an $m$-edge graph, there exists no dynamic algorithms with both $O(m^{1/2 - \epsilon})$ update time and $ O(m^{1 -\epsilon})$ query time, for any small $\epsilon > 0$. Note that for ($s,t$)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and $O(m)$ query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching.