A Linear-time Algorithm for Integral Multiterminal Flows in Trees

TL;DR

This paper presents a linear-time algorithm for finding maximum integral multiterminal flows and corresponding minimum cut-systems in trees, significantly improving efficiency over previous methods.

Contribution

It introduces an $O(n)$-time dynamic programming algorithm that computes maximum integral multiterminal flows and minimum cut-systems in trees, enhancing previous results by a factor of $ olinebreak ext{O}(n)$.

Findings

Algorithm runs in linear time $O(n)$ for trees.

Constructs maximum integral multiterminal flows and minimum cut-systems efficiently.

Improves previous algorithms' complexity by a factor of $ ext{O}(n)$.

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.