Improved Analysis of EDCS via Gallai-Edmonds Decomposition

TL;DR

This paper improves the theoretical bounds on the edge-degree constrained subgraph (EDCS) for approximating maximum matchings in general graphs by using Gallai-Edmonds decomposition, removing previous gaps in understanding.

Contribution

It provides a new proof of EDCS's approximation guarantee using Gallai-Edmonds decomposition, achieving tighter bounds for general graphs.

Findings

Achieves a $(2/3 - \epsilon)$-approximation with EDCS of maximum degree $O(1/\epsilon)$

Improves previous bounds from $O(rac{ ext{log}(1/\epsilon)}{\epsilon^2})$ and $O(1/\epsilon^3)$

Bridges the gap between bipartite and general graphs in EDCS analysis.

Abstract

In this note, we revisit the edge-degree constrained subgraph (EDCS) introduced by Bernstein and Stein (ICALP'15). An EDCS is a sparse subgraph satisfying simple edge-degree constraints that is guaranteed to include an (almost) $\frac{2}{3}$-approximate matching of the base graph. Since its introduction, the EDCS has been successfully applied to numerous models of computation. Motivated by this success, we revisit EDCS and present an improved bound for its key property in general graphs. Our main result is a new proof of the approximation guarantee of EDCS that builds on the graph's Gallai-Edmonds decomposition, avoiding the probabilistic method of the previous proofs. As a result, we get that to obtain a $(\frac{2}{3} - \epsilon)$-approximation, a sparse EDCS with maximum degree bounded by $O(1/\epsilon)$ is sufficient. This improves the $O(\log(1/\epsilon)/\epsilon^2)$ bound of Assadi and Bernstein (SOSA'19) and the $O(1/\epsilon^3)$ bound of Bernstein and Stein (SODA'16). Our guarantee essentially matches what was previously only known for bipartite graphs, thereby removing the gap in our understanding of EDCS in general vs. bipartite graphs.