A Cost-Scaling Algorithm for Minimum-Cost Node-Capacitated Multiflow Problem

In this paper, we address the minimum-cost node-capacitated multiflow problem in an undirected network. For this problem, Babenko and Karzanov (2012) showed strongly polynomial-time solvability via the ellipsoid method. Our result is the first combinatorial weakly 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 $\mathrm{SF}(n', m', η)$ is the time complexity of solving the submodular flow problem in a network of $n'$ nodes, $m'$ edges, and a submodular function with $η$-time-computable exchange capacity. Our algorithm is built on discrete convex analysis on graph structures and the concept of reducible bisubmodular flows.