  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].

Discrete Mathematics