A cost-scaling algorithm for minimum-cost node-capacitated multiflow problem

Abstract

In this paper, we address the minimum-cost node-capacitated multiflow problem in undirected networks. For this problem, Babenko and Karzanov (JCO 24: 202–228, 2012) showed strong polynomial-time solvability via the ellipsoid method. Our result is the first combinatorial polynomial-time algorithm for this problem. Our algorithm finds a half-integral minimum-cost maximum multiflow in \(O(m \log (nCD)\mathrm {SF}(kn,m,k))\) time, where n is the number of nodes, m is the number of edges, k is the number of terminals, C is the maximum node capacity, D is the maximum edge cost, and \(\mathrm {SF}(n',m',\eta )\) is the time complexity of solving the submodular flow problem in a network of \(n'\) nodes, \(m'\) edges, and a submodular function with \(\eta \)-time-computable exchange capacity. Our algorithm is built on discrete convex analysis on graph structures and the concept of reducible bisubmodular flows.