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# A Linear-Time Algorithm for Integral Multiterminal Flows in Trees

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## Cite As

Mingyu Xiao and Hiroshi Nagamochi. A Linear-Time Algorithm for Integral Multiterminal Flows in Trees. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 62:1-62:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ISAAC.2016.62

## Abstract

In this paper, we study the problem of finding an integral multiflow which maximizes the sum of flow values between every two terminals in an undirected tree with a nonnegative integer edge capacity and a set of terminals. In general, it is known that the flow value of an integral multiflow is bounded by the cut value of a cut-system which consists of disjoint subsets each of which contains exactly one terminal or has an odd cut value, and there exists a pair of an integral multiflow and a cut-system whose flow value and cut value are equal; i.e., a pair of a maximum integral multiflow and a minimum cut. In this paper, we propose an O(n)-time algorithm that finds such a pair of an integral multiflow and a cut-system in a given tree instance with n vertices. This improves the best previous results by a factor of Omega(n). Regarding a given tree in an instance as a rooted tree, we define O(n) rooted tree instances taking each vertex as a root, and establish a recursive formula on maximum integral multiflow values of these instances to design a dynamic programming that computes the maximum integral multiflow values of all O(n) rooted instances in linear time. We can prove that the algorithm implicitly maintains a cut-system so that not only a maximum integral multiflow but also a minimum cut-system can be constructed in linear time for any rooted instance whenever it is necessary. The resulting algorithm is rather compact and succinct.
##### Keywords
• Multiterminal flow; Maximum flow; Minimum Cut; Trees; Linear-time algorithms

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## References

1. B. V. Cherkasskii. Reshenie odnoi zadachi o mnogoproduktovykh potokakh v seti [russian; a solution of a problem of multicommodity flows in a network]. Èkonomika i Matematicheskie Metody, 13(1):143-151, 1977.
2. Marie-Christine Costa, Lucas Létocart, and Frédéric Roupin. Minimal multicut and maximal integer multiflow: a survey. European Journal of Operational Research, 162(1):55-69, 2005.
3. Mariechristine Costa and Alain Billionnet. Multiway cut and integer flow problems in trees. Electronic Notes in Discrete Mathematics, 17(20):105-109, 2004.
4. William H. Cunningham. The optimal multiterminal cut problem. DIMACS series in discrete mathematics and theoretical computer science, 5:105-120, 1991.
5. J. R. Ford and D. R. Fulkerson. Flows in networks. Princeton university press, 1962.
6. Naveen Garg, Vijay V. Vazirani, and Mihalis Yannakakis. Approximate max-flow min-(multi) cut theorems and their applications. SIAM Journal on Computing, 25(2):235-251, 1996.
7. Naveen Garg, Vijay V. Vazirani, and Mihalis Yannakakis. Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica, 18(1):3-20, 1997.
8. Torben Hagerup, Jyrki Katajainen, Naomi Nishimura, and Prabhakar Ragde. Characterizing multiterminal flow networks and computing flows in networks of small treewidth. Journal of Computer and System Sciences, 57(3):366-375, 1998.
9. Toshihide Ibaraki, Alexander V. Karzanov, and Hiroshi Nagamochi. A fast algorithm for finding a maximum free multiflow in an inner eulerian network and some generalizations. Combinatorica, 18(1):61-83, 1998.
10. Wolfgang Mader. Über die Maximalzahl kantendisjunkter A-Wege. Archiv der Mathematik, 30(1):325-336, 1978.
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