Simple linear flow decomposition algorithms on trees, circles, and augmented trees

Abstract

The flow decomposition algorithm transforms an arc flow-based solution to a network flow problem into flows on directed paths and cycles. When the undirected graph induced by arcs with positive flow is a tree, a circle, or an augmented tree (with n nodes), the standard implementation of the algorithm runs in O (n 2) time. In this article, we exploit the structure of the network to develop an O (n) flow decomposition algorithm. The run-time relies on the property that for these networks, paths or cycles can be represented implicitly in O (1) space. The algorithm is easy to implement and does not use complicated data structures. Because the size of the input is O (n), our algorithm is the fastest possible for flow decomposition on these special networks. Our algorithm can be used to improve run-times for solving matching and transportation problems on trees and circles. © 2012 Wiley Periodicals, Inc.