A dual descent algorithm for node-capacitated multiflow problems and its applications

TL;DR

This paper introduces a new strongly polynomial combinatorial algorithm for maximum node-capacitated multiflow problems, leveraging discrete convex analysis and submodular flow techniques, with applications to node multiway cut approximations.

Contribution

It presents the first combinatorial strongly polynomial algorithm for maximum node-capacitated multiflow, utilizing discrete convex functions and submodular flow methods.

Findings

Achieves $O((m \log k) ext{MSF}(n,m,1))$ time complexity.

Finds maximum half-integral multiflow in $O(m n^3 \log k)$ time.

Enables ellipsoid-free 2-approximation for minimum node multiway cut.

Abstract

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.