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Using the minimum maximum flow degree to approximate the flow coloring problem

  • S.I.: CLAIO 2018
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A Correction to this article was published on 20 April 2022

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Abstract

Consider an arc-capacitated network \(\mathcal {N}\) through which an integer-valued flow must be sent from several source nodes to a sink node. Each feasible flow defines a corresponding multigraph with the same vertices as \(\mathcal {N}\) and an edge for each arc of \(\mathcal {N}\), where the edge multiplicity is the flow in the respective arc. The maximum flow degree of a feasible flow is the maximum sum of the flow entering and leaving a node of \(\mathcal {N}\), i.e. the maximum degree of the corresponding multigraph. The minimum maximum flow degree problem (MMFDP) consists in determining on \(\mathcal {N}\) a feasible flow such that its maximum flow degree is minimum. We present a polynomial time algorithm for this problem. We use its optimum value to derive an improved upper bound for the flow coloring problem (FCP), which consists in finding a feasible flow whose corresponding multigraph has the minimum chromatic index. Based on this procedure, we design an approximation algorithm for the FCP that improves the best known approximation factor.

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Notes

  1. The FCP has been defined on an undirected graph \(G=(V,E)\), with a demand function \(b:V\rightarrow \mathbb {Z}_+\), where a flow from the sources to t can pass through the edges in any direction with no capacity limitation. Thus, associated with G, there is a network \(\mathcal {N}=(V,A,b,c\equiv \infty )\) with two arcs in A with opposite directions for each edge of G. Here, we preferred to define FCP directly on \(\mathcal {N}\) to emphasis its relation with MMFDP.

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Authors and Affiliations

  1. Departamento de Estatística e Matemática Aplicada, Universidade Federal do Ceará, Fortaleza, Brasil

    Manoel Campêlo

  2. Programa de Mestrado e Doutorado em Ciência da Computação, Universidade Federal do Ceará, Fortaleza, Brasil

    Jhonata A. S. Matias

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  1. Manoel Campêlo
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  2. Jhonata A. S. Matias
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Correspondence to Jhonata A. S. Matias.

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M. Campêlo: Partially supported by CNPq, Brazil (425297/2016-0, 309315/2019-0), and FUNCAP-CE/Brazil (PNE-0112-00061.01.00/16). J. A. S. Matias: Supported by FUNCAP-CE/Brazil.

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Campêlo, M., Matias, J.A.S. Using the minimum maximum flow degree to approximate the flow coloring problem. Ann Oper Res 316, 1267–1278 (2022). https://doi.org/10.1007/s10479-021-04180-3

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  • DOI: https://doi.org/10.1007/s10479-021-04180-3

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