Optimal Online Bipartite Matching in Degree-2 Graphs

TL;DR

This paper investigates online bipartite matching in degree-2 graphs, revealing a separation between fractional and integral matchings and establishing optimal competitive ratios for these classes.

Contribution

It proves the optimality of the Half-Half algorithm for fractional matchings in degree-2 graphs and demonstrates a fundamental difference between fractional and integral online matchings.

Findings

Half-Half algorithm achieves approximately 0.71777 competitive ratio

Optimal fractional matching ratio is 0.75 in degree-2 graphs

No perfect rounding scheme exists for these matchings

Abstract

Online bipartite matching is a classical problem in online algorithms and we know that both the deterministic fractional and randomized integral online matchings achieve the same competitive ratio of $1-\frac{1}{e}$. In this work, we study classes of graphs where the online degree is restricted to $2$. As expected, one can achieve a competitive ratio of better than $1-\frac{1}{e}$ in both the deterministic fractional and randomized integral cases, but surprisingly, these ratios are not the same. It was already known that for fractional matching, a $0.75$ competitive ratio algorithm is optimal. We show that the folklore \textsc{Half-Half} algorithm achieves a competitive ratio of $\eta \approx 0.717772\dots$ and more surprisingly, show that this is optimal by giving a matching lower-bound. This yields a separation between the two problems: deterministic fractional and randomized integral, showing that it is impossible to obtain a perfect rounding scheme.