Tight Competitive Ratios of Classic Matching Algorithms in the Fully Online Model

TL;DR

This paper determines the exact competitive ratios of classic algorithms for the fully online matching problem, establishing tight bounds for both fractional and integral cases and improving existing hardness results.

Contribution

It provides the first tight competitive ratio bounds for the fractional water-filling algorithm and the ranking algorithm in the fully online matching model.

Findings

Fractional water-filling algorithm is 2−√2≈0.585-competitive.

Ranking algorithm on bipartite graphs has a tight ratio of approximately 0.567.

Hardness results improve previous upper bounds for the problem.

Abstract

Huang et al.~(STOC 2018) introduced the fully online matching problem, a generalization of the classic online bipartite matching problem in that it allows all vertices to arrive online and considers general graphs. They showed that the ranking algorithm by Karp et al.~(STOC 1990) is strictly better than $0.5$-competitive and the problem is strictly harder than the online bipartite matching problem in that no algorithms can be $(1-1/e)$-competitive. This paper pins down two tight competitive ratios of classic algorithms for the fully online matching problem. For the fractional version of the problem, we show that a natural instantiation of the water-filling algorithm is $2-\sqrt{2} \approx 0.585$-competitive, together with a matching hardness result. Interestingly, our hardness result applies to arbitrary algorithms in the edge-arrival models of the online matching problem, improving the state-of-art $\frac{1}{1+\ln 2} \approx 0.5906$ upper bound. For integral algorithms, we show a tight competitive ratio of $\approx 0.567$ for the ranking algorithm on bipartite graphs, matching a hardness result by Huang et al. (STOC 2018).