# On adaptive algorithms for maximum matching

## TL;DR

This paper demonstrates that phase-based maximum matching algorithms are adaptive to input structure, achieving linear or near-linear time on certain graph classes and graphs close to these classes, while maintaining worst-case bounds.

## Contribution

It shows that phase-based algorithms adaptively run faster on structured graphs and graphs close to such classes, without explicit deletion set computation.

## Key findings

- Algorithms run in linear time on specific graph classes.
- Algorithms run in
O(\u222a k m) for graphs close to these classes.
- Phase framework can still require
(    n m) time on simple graph classes.

## Abstract

In the fundamental Maximum Matching problem the task is to find a maximum cardinality set of pairwise disjoint edges in a given undirected graph. The fastest algorithm for this problem, due to Micali and Vazirani, runs in time $\mathcal{O}(\sqrt{n}m)$ and stands unbeaten since 1980. It is complemented by faster, often linear-time, algorithms for various special graph classes. Moreover, there are fast parameterized algorithms, e.g., time $\mathcal{O}(km\log n)$ relative to tree-width $k$, which outperform $\mathcal{O}(\sqrt{n}m)$ when the parameter is sufficiently small. We show that the Micali-Vazirani algorithm, and in fact any algorithm following the phase framework of Hopcroft and Karp, is adaptive to beneficial input structure. We exhibit several graph classes for which such algorithms run in linear time $\mathcal{O}(n+m)$. More strongly, we show that they run in time $\mathcal{O}(\sqrt{k}m)$ for graphs that are $k$ vertex deletions away from any of several such classes, without explicitly computing an optimal or approximate deletion set; before, most such bounds were at least $\Omega(km)$. Thus, any phase-based matching algorithm with linear-time phases obliviously interpolates between linear time for $k=\mathcal{O}(1)$ and the worst case of $\mathcal{O}(\sqrt{n}m)$ when $k=\Theta(n)$. We complement our findings by proving that the phase framework by itself still allows $\Omega(\sqrt{n})$ phases, and hence time $\Omega(\sqrt{n}m)$, even on paths, cographs, and bipartite chain graphs.