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. [7] 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 [5].
The work of all authors was supported by Polish National Science Center grant 2013/11/D/ST6/03100. Additionally, the work of P. Sankowski was partially supported by the project TOTAL (No. 677651) that has received funding from ERC.
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Notes
- 1.
If there are more such vertices we choose the first one according to some predefined order on \(B\cup W\).
- 2.
Symbol \(\cdot \) denotes concatenation of paths.
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Bosek, B., Leniowski, D., Sankowski, P., Zych-Pawlewicz, A. (2018). A Tight Bound for Shortest Augmenting Paths on Trees. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_16
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