Edge Arrival Online Matching: The Power of Free Disposal on Acyclic Graphs

TL;DR

This paper investigates the impact of free disposal in online matching on acyclic graphs, demonstrating improved competitive ratios in unweighted and weighted scenarios, and establishing the power of free disposal in these settings.

Contribution

It provides the first evidence of free disposal enhancing competitive ratios in unweighted online matching on Growing Trees and Forests, and improves ratios for weighted matching with a simple optimal algorithm.

Findings

Improved unweighted matching ratio on Growing Trees from 5/9 to 2/3.

Enhanced unweighted matching ratio on Forests from 5/9 to 5/8.

Weighted matching ratio improved from 1/3 to 1/2 with an optimal ordinal algorithm.

Abstract

Online matching is a fundamental problem in the study of online algorithms. We study the problem under a very general arrival model: the edge arrival model. Free disposal is an important notion in the online matching literature, which allows the algorithm to dispose of the previously matched edges. Without free disposal, we cannot achieve any bounded ratio, even with randomized algorithms, when edges are weighted. Our paper focuses on clarifying the power of free disposal in both the unweighted and the weighted setting. As far as we know, it's still uncertain if free disposal can give us extra leverage to enhance the competitive ratio in the unweighted scenario, even in specific instances such as Growing Trees, where every new edge adds a new leaf to the graph. Our study serves as a valuable initial exploration of this open question. The results are listed as follows: 1. With free disposal, we improve the competitive ratio for unweighted online matching on Growing Trees from $5/9$ to $2/3 \approx 0.66$, and show that the ratio is tight. For Forests, a more general setting where the underlying graph is a forest and edges may arrive in arbitrary order, we improve the competitive ratio from $5/9$ to $5/8 = 0.625$. 2. Both the ratios of $2/3$ and $0.625$ show a separation to the upper bound of the competitive ratio without free disposal on Growing Trees ($0.5914$). Therefore, we demonstrate the additional power of free disposal for the unweighted setting for the first time, at least in the special setting of Growing Trees and Forests. 3. We improve the competitive ratio for weighted online matching on Growing Trees from $1/3$ to $1/2$ using a very simple ordinal algorithm, and show that it is optimal among ordinal algorithms.