  In this paper, we develop an $O((m \log k) {\rm MSF} (n,m,1))$-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
${\rm MSF} (n',m',\gamma)$ denotes the time complexity of solving the maximum
submodular flow problem in a network with $n'$ nodes, $m'$ edges, and the
complexity $\gamma$ 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.
