### Editor: Seok-Hee Hong; Article No. 62

Mingyu Xiao, Hiroshi Nagamochi
2016 12 Leibniz International Proceedings in Informatics Schloss Dagstuhl-Leibniz-Zentrum für Informatik   unpublished
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
more » ... ultiflow 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 Ω(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. 1998 ACM Subject Classification G.2.2 Graph Theory The min-cut max-flow theorem by Ford and Fulkerson  is one of the most important theorems in graph theory. It catches a min-max relation between two fundamental graph problems. This theorem leads to many effective algorithms and much theory for flow problems as well as graph cut problems. Due to the great applications of it, researchers have interests to seek more similar min-max formulas in various kinds of flow and cut problems. In this paper, we consider the maximum multiterminal flow problem, a generalization of the basic maximum flow problem. * A full version of the paper is available at https://arxiv.org/abs/1611.08803.