{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9238","name":"A Linear-Time Algorithm for Integral Multiterminal Flows in Trees","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","author":[{"@type":"Person","name":"Xiao, Mingyu","givenName":"Mingyu","familyName":"Xiao"},{"@type":"Person","name":"Nagamochi, Hiroshi","givenName":"Hiroshi","familyName":"Nagamochi"}],"position":62,"pageStart":"62:1","pageEnd":"62:12","dateCreated":"2016-12-07","datePublished":"2016-12-07","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Xiao, Mingyu","givenName":"Mingyu","familyName":"Xiao"},{"@type":"Person","name":"Nagamochi, Hiroshi","givenName":"Hiroshi","familyName":"Nagamochi"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ISAAC.2016.62","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6267","volumeNumber":64,"name":"27th International Symposium on Algorithms and Computation (ISAAC 2016)","dateCreated":"2016-12-07","datePublished":"2016-12-07","editor":{"@type":"Person","name":"Hong, Seok-Hee","givenName":"Seok-Hee","familyName":"Hong"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9238","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6267"}}}