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Enumerating k-Arc-Connected Orientations

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Abstract

We study the problem of enumerating the k-arc-connected orientations of a graph G, i.e., generating each exactly once. A first algorithm using submodular flow optimization is easy to state, but intricate to implement. In a second approach we present a simple algorithm with \(O(knm^2)\) time delay and amortized time \(O(m^2)\), which improves over the analysis of the submodular flow algorithm. As ingredients, we obtain enumeration algorithms for the \(\alpha \)-orientations of a graph G in \(O(m^2)\) time delay and for the outdegree sequences attained by k-arc-connected orientations of G in \(O(knm^2)\) time delay.

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Notes

  1. We use the term enumeration instead of the sometimes used terms generation or listing.

  2. When there is no ambiguity, we will abbreviate k-arc-connected by saying k-connected.

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Acknowledgements

We wish to thank Nadia Creignou and Frédéric Olive for fruitful discussions in an early stage of this paper. A preliminary version of the results obtained in this work was presented at WEPA 2018: Second Workshop on Enumeration Problems and Applications which was held in Pisa on November 2018. The authors wish to thank the organizers of this workshop.

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Authors and Affiliations

  1. LGIPM, Université de Lorraine, 57000, Metz, France

    Sarah Blind

  2. Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Barcelona, Spain

    Kolja Knauer

  3. LIS - Aix-Marseille Univ, CNRS, Universié de Toulon, Marseille, France

    Kolja Knauer & Petru Valicov

  4. LIRMM - CNRS, Université de Montpellier, Montpellier, France

    Petru Valicov

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  1. Sarah Blind
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  2. Kolja Knauer
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  3. Petru Valicov
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Correspondence to Petru Valicov.

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The first author was supported by the French ANR project GraphEn: ANR-15-CE40-0009. The second and third authors were supported by the French ANR project DISTANCIA: ANR-17-CE40-0015. The second author moreover was supported by ANR grant GATO: ANR-16-CE40-0009-01 and by the Spanish Ministerio de Economía, Industria y Competitividad through Grant RYC-2017-22701.

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Blind, S., Knauer, K. & Valicov, P. Enumerating k-Arc-Connected Orientations. Algorithmica 82, 3588–3603 (2020). https://doi.org/10.1007/s00453-020-00738-y

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