# A Tight Bound for Shortest Augmenting Paths on Trees

## TL;DR

This paper proves a tight upper bound of O(n log n) for the total length of shortest augmenting paths in online bipartite matching on trees, improving previous bounds and deepening understanding of this fundamental technique.

## Contribution

It establishes a tight O(n log n) bound on shortest augmenting paths in online bipartite matchings on trees, resolving a conjecture and improving prior results.

## Key findings

- Proves a tight O(n log n) bound for augmenting paths
- Improves over previous O(n log^2 n) bound
- Deepens theoretical understanding of online bipartite matching on trees

## Abstract

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree $T=(W \uplus B, E)$ is being revealed online, i.e., in each round one vertex from $B$ with its incident edges arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis, R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with augmentations. In INFOCOM 2009] that the total length of all shortest augmenting paths found is $O(n \log n)$. In this paper, we prove a tight $O(n \log n)$ upper bound for the total length of shortest augmenting paths for trees improving over $O(n \log^2 n)$ bound [B. Bosek, D. Leniowski, P. Sankowski, and A. Zych. Shortest augmenting paths for online matchings on trees. In WAOA 2015].