Improved Approximation for Ranking on General Graphs

TL;DR

This paper improves the approximation ratio of the Ranking algorithm for general graphs from 0.526 to 0.5469, using new structural insights and primal-dual analysis, surpassing previous bounds and advancing understanding of randomized greedy matchings.

Contribution

The authors introduce the concept of vertex backups to analyze the Ranking algorithm, leading to a higher approximation ratio for general graphs.

Findings

Approximation ratio improved to 0.5469 for general graphs.

New structural properties of Ranking are established using vertex backups.

The approach surpasses previous best bounds and enhances theoretical understanding.

Abstract

In this paper, we study Ranking, a well-known randomized greedy matching algorithm, for general graphs. The algorithm was originally introduced by Karp, Vazirani, and Vazirani [STOC 1990] for the online bipartite matching problem with one-sided vertex arrivals, where it achieves a tight approximation ratio of 1 - 1/e. It was later extended to general graphs by Goel and Tripathi [FOCS 2012]. The Ranking algorithm for general graphs is as follows: a permutation $\sigma$ over the vertices is chosen uniformly at random. The vertices are then processed sequentially according to this order, with each vertex being matched to the first available neighbor (if any) according to the same permutation $\sigma$. While the algorithm is quite well-understood for bipartite graphs-with the approximation ratio lying between 0.696 and 0.727, its approximation ratio for general graphs remains less well characterized despite extensive efforts. Prior to this work, the best known lower bound for general graphs was 0.526 by Chan et al. [TALG 2018], improving on the approximation ratio of 0.523 by Chan et al. [SICOMP 2018]. The upper bound, however, remains the same as that for bipartite graphs. In this work, we improve the approximation ratio of \textsc{Ranking} for general graphs to 0.5469, up from 0.526. This also surpasses the best-known approximation ratio of $0.531$ by Tang et al. [JACM 2023] for the oblivious matching problem. Our approach builds on the standard primal-dual analysis. The novelty of our work lies in proving new structural properties of Ranking by introducing the notion of the backup for vertices matched by the algorithm. For a fixed permutation, a vertex's backup is its potential match if its current match is removed. This concept helps characterize the rank distribution of the match of each vertex, enabling us to eliminate certain bad events that constrained previous work.