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

TL;DR

This paper introduces new classes of discrete convex functions on trees, develops a theory with optimization properties, and applies it to create a polynomial proximity scaling algorithm for the minimum cost multiflow problem, improving computational efficiency.

Contribution

The paper develops a novel theory of L-extendable and alternating L-convex functions, and applies it to design the first polynomial-time combinatorial algorithm for the minimum cost multiflow problem.

Findings

Introduced L-extendable and alternating L-convex functions with key properties.

Developed a polynomial proximity scaling algorithm for the minimum cost multiflow problem.

Extended the approach to solve the more general node-demand multiflow problem.

Abstract

In this paper, we develop a theory of new classes of discrete convex functions, called L-extendable functions and alternating L-convex functions, defined on the product of trees. We establish basic properties for optimization: a local-to-global optimality criterion, the steepest descend algorithm by successive $k$-submodular function minimizations, the persistency property, and the proximity theorem. Our theory is motivated by minimum cost free multiflow problem. To this problem, Goldberg and Karzanov gave two combinatorial weakly polynomial time algorithms based on capacity and cost scalings, without explicit running time. As an application of our theory, we present a new simple polynomial proximity scaling algorithm to solve minimum cost 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 of terminals, $A$ is the maximum of edge-costs, $C$ is the total sum of edge-capacities, and ${\rm MF}(n',m')$ denotes the time complexity to find a maximum flow in a network of $n'$ nodes and $m'$ edges. Our algorithm is designed to solve, in the same time complexity, a more general class of multiflow problems, minimum cost node-demand multiflow problem, and is the first combinatorial polynomial time algorithm to this class of problems. We also give an application to network design problem.