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.

在本文中，我们解决了无向网络中最小成本节点容量的多流问题。对于这个问题，Babenko 和 Karzanov (JCO 24: 202–228, 2012) 通过椭球法显示了很强的多项式时间可解性。我们的结果是这个问题的第一个组合多项式时间算法。我们的算法在\(O(m \log (nCD)\mathrm {SF}(kn,m,k))\)时间内找到一个半积分最小成本最大多流，其中n是节点数，m是边数，k为终端数，C为最大节点容量，D为最大边成本，\(\mathrm {SF}(n',m',\eta )\)是在\(n'\)节点、\(m'\)边和具有\(\eta \)时间可计算交换容量的子模函数的网络中解决子模流问题的时间复杂度。我们的算法建立在图结构的离散凸分析和可约双子模流的概念之上。