Fully Online Matching II: Beating Ranking and Water-filling

TL;DR

This paper introduces new algorithms for fully online matching that outperform previous methods like Ranking and Water-filling, achieving higher competitive ratios for both integral and fractional matchings in general graphs.

Contribution

It presents the first algorithms that surpass Ranking and Water-filling in the fully online matching model, including a hybrid integral matching algorithm and an eager fractional matching algorithm.

Findings

Balanced Ranking achieves 0.569-competitiveness for integral matching.

Eager Water-filling achieves 0.592-competitiveness for fractional matching.

Results demonstrate a separation from vertex arrival models with lower bounds of 0.5914.

Abstract

Karp, Vazirani, and Vazirani (STOC 1990) initiated the study of online bipartite matching, which has held a central role in online algorithms ever since. Of particular importance are the Ranking algorithm for integral matching and the Water-filling algorithm for fractional matching. Most algorithms in the literature can be viewed as adaptations of these two in the corresponding models. Recently, Huang et al.~(STOC 2018, SODA 2019) introduced a more general model called \emph{fully online matching}, which considers general graphs and allows all vertices to arrive online. They also generalized Ranking and Water-filling to fully online matching and gave some tight analysis: Ranking is $\Omega \approx 0.567$-competitive on bipartite graphs where the $\Omega$-constant satisfies $\Omega e^\Omega = 1$, and Water-filling is $2-\sqrt{2} \approx 0.585$-competitive on general graphs. We propose fully online matching algorithms strictly better than Ranking and Water-filling. For integral matching on bipartite graphs, we build on the online primal dual analysis of Ranking and Water-filling to design a $0.569$-competitive hybrid algorithm called Balanced Ranking. To our knowledge, it is the first integral algorithm in the online matching literature that successfully integrates ideas from Water-filling. For fractional matching on general graphs, we give a $0.592$-competitive algorithm called Eager Water-filling, which may match a vertex on its arrival. By contrast, the original Water-filling algorithm always matches vertices at their deadlines. Our result for fractional matching further shows a separation between fully online matching and the general vertex arrival model by Wang and Wong (ICALP 2015), due to an upper bound of $0.5914$ in the latter model by Buchbinder, Segev, and Tkach (ESA 2017).