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A cost-scaling algorithm for minimum-cost node-capacitated multiflow problem

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Abstract

In this paper, we address the minimum-cost node-capacitated multiflow problem in undirected networks. For this problem, Babenko and Karzanov (JCO 24: 202–228, 2012) showed strong polynomial-time solvability via the ellipsoid method. Our result is the first combinatorial polynomial-time algorithm for this problem. Our algorithm finds a half-integral minimum-cost maximum multiflow in \(O(m \log (nCD)\mathrm {SF}(kn,m,k))\) time, where n is the number of nodes, m is the number of edges, k is the number of terminals, C is the maximum node capacity, D is the maximum edge cost, and \(\mathrm {SF}(n',m',\eta )\) is the time complexity of solving the submodular flow problem in a network of \(n'\) nodes, \(m'\) edges, and a submodular function with \(\eta \)-time-computable exchange capacity. Our algorithm is built on discrete convex analysis on graph structures and the concept of reducible bisubmodular flows.

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Notes

  1. Actually, in [1], these facts are shown only for the case \(\underline{b}=0\). But the same proofs work in the case \(\underline{b}\ne 0\).

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Acknowledgements

We thank referees for many useful comments and suggestions for improving the quality of the paper. The first author was supported by JSPS KAKENHI Grant Numbers JP17K00029 and JST PRESTO Grant Number JPMJPR192A, Japan.

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  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, 113-8656, Japan

    Hiroshi Hirai & Motoki Ikeda

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  1. Hiroshi Hirai
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  2. Motoki Ikeda
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Correspondence to Hiroshi Hirai.

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A preliminary version of this paper appears in the proceeding of the 11th Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications, 2019.

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Hirai, H., Ikeda, M. A cost-scaling algorithm for minimum-cost node-capacitated multiflow problem. Math. Program. 195, 149–181 (2022). https://doi.org/10.1007/s10107-021-01683-6

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