In this article, we develop an O((m log k)MSF(n,m,1))-time algorithm to find a half-integral node-capacitated multiflow of the maximum total flow-value in a network with n nodes, m edges, and k terminals, where MSF(n^{′},m^{′},γ) denotes the time complexity of solving the maximum submodular flow problem in a network with n^{′} nodes, m^{′} edges, and the complexity γ of computing the exchange capacity of the submodular function describing the problem. By using Fujishige-Zhang algorithm for submodular flow, we can find a maximum half-integral multiflow in O(m n^{3} log k) time. This is the first combinatorial strongly polynomial time algorithm for this problem. Our algorithm is built on a developing theory of discrete convex functions on certain graph structures. Applications include “ellipsoid-free” combinatorial implementations of a 2-approximation algorithm for the minimum node-multiway cut problem by Garg, Vazirani, and Yannakakis.