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In this paper, we develop a theory of new classes of discrete convexfunctions, called L-extendable functions and alternating L-convex functions,defined on the product of trees. We establish basic properties foroptimization: a local-to-global optimality criterion, the steepest descendalgorithm by successive $k$-submodular function minimizations, the persistencyproperty, and the proximity theorem. Our theory is motivated by minimum costfree multiflow problem. To this problem, Goldberg and Karzanov gave twocombinatorial weakly polynomial time algorithms based on capacity and costscalings, without explicit running time. As an application of our theory, wepresent a new simple polynomial proximity scaling algorithm to solve minimumcost free multiflow problem in $O(n \log (n AC) {\rm MF}(kn, km))$ time, where$n$ is the number of nodes, $m$ is the number of edges, $k$ is the number ofterminals, $A$ is the maximum of edge-costs, $C$ is the total sum ofedge-capacities, and ${\rm MF}(n',m')$ denotes the time complexity to find amaximum flow in a network of $n'$ nodes and $m'$ edges. Our algorithm isdesigned to solve, in the same time complexity, a more general class ofmultiflow problems, minimum cost node-demand multiflow problem, and is thefirst combinatorial polynomial time algorithm to this class of problems. Wealso give an application to network design problem.

L-extendable functions and a proximity scaling algorithm for minimum cost multiflow problem