Bipartite Matching in Massive Graphs: A Tight Analysis of EDCS

TL;DR

This paper provides a tight analysis of the edge-degree constrained subgraph (EDCS) sparsifier for maximum matching in massive graphs, revealing that a specific parameter choice yields an approximation ratio better than previously believed.

Contribution

The authors derive a tight, parameter-specific approximation ratio for EDCS, showing that the best ratio is achieved at b2=6, surpassing the long-held 2/3 bound.

Findings

Optimal b2=6 yields a 0.677 approximation ratio.

Previous bounds suggested increasing b2 improves approximation.

The analysis is tight for all b2 values.

Abstract

Maximum matching is one of the most fundamental combinatorial optimization problems with applications in various contexts such as balanced clustering, data mining, resource allocation, and online advertisement. In many of these applications, the input graph is massive. The sheer size of these inputs makes it impossible to store the whole graph in the memory of a single machine and process it there. Graph sparsification has been an extremely powerful tool to alleviate this problem. In this paper, we study a highly successful and versatile sparsifier for the matching problem: the *edge-degree constrained subgraph (EDCS)* introduced first by Bernstein and Stein [ICALP'15]. The EDCS has a parameter $\beta \geq 2$ which controls the density of the sparsifier. It has been shown through various proofs in the literature that by picking a subgraph with $O(n\beta)$ edges, the EDCS includes a matching of size at least $2/3-O(1/\beta)$ times the maximum matching size. As such, by increasing $\beta$ the approximation ratio of EDCS gets closer and closer to $2/3$. In this paper, we propose a new approach for analyzing the approximation ratio of EDCS. Our analysis is *tight* for any value of $\beta$. Namely, we pinpoint the precise approximation ratio of EDCS for any sparsity parameter $\beta$. Our analysis reveals that one does not necessarily need to increase $\beta$ to improve approximation, as suggested by previous analysis. In particular, the best choice turns out to be $\beta = 6$, which achieves an approximation ratio of $.677$! This is arguably surprising as it is even better than $2/3 \sim .666$, the bound that was widely believed to be the limit for EDCS.