New Trade-Offs for Fully Dynamic Matching via Hierarchical EDCS

TL;DR

This paper introduces a hierarchical EDCS technique to improve the trade-offs in fully dynamic approximate maximum matching algorithms, unifying and extending previous approaches.

Contribution

It presents a new hierarchical EDCS method that generalizes existing techniques, revealing new trade-offs in dynamic matching algorithms.

Findings

Unifies previous trade-offs in dynamic matching algorithms

Introduces hierarchical EDCS as a new tool

Achieves improved understanding of the trade-off spectrum

Abstract

We study the maximum matching problem in fully dynamic graphs: a graph is undergoing both edge insertions and deletions, and the goal is to efficiently maintain a large matching after each edge update. This problem has received considerable attention in recent years. The known algorithms naturally exhibit a trade-off between the quality of the matching maintained (i.e., the approximation ratio) and the time needed per update. While several interesting results have been obtained, the optimal behavior of this trade-off remains largely unclear. Our main contribution is a new approach to designing fully dynamic approximate matching algorithms that in a unified manner not only (essentially) recovers all previously known trade-offs that were achieved via very different techniques, but reveals some new ones as well. As our main tool to achieve this, we introduce a generalization of the edge-degree constrained subgraph (EDCS) of Bernstein and Stein (2015) that we call the hierarchical EDCS (HEDCS).