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Degree-Constrained Orientation of Maximum Satisfaction: Graph Classes and Parameterized Complexity

Abstract

The problem Max W-Light (Max W-Heavy) for an undirected graph is to assign a direction to each edge so that the number of vertices of outdegree at most W (resp. at least W) is maximized. It is known that these problems are NP-hard even for fixed W. For example, Max 0-Light is equivalent to the problem of finding a maximum independent set. In this paper, we show that for any fixed constant W, Max W-Heavy can be solved in linear time for hereditary graph classes for which treewidth is bounded by a function of degeneracy. We show that such graph classes include chordal graphs, circular-arc graphs, d-trapezoid graphs, chordal bipartite graphs, and graphs of bounded clique-width. To have a polynomial-time algorithm for Max W-Light, we need an additional condition of a polynomial upper bound on the number of potential maximal cliques to apply the metatheorem by Fomin et al. (SIAM J Comput 44:54–87, 2015). The aforementioned graph classes, except bounded clique-width graphs, satisfy such a condition. For graphs of bounded clique-width, we present a dynamic programming approach not using the metatheorem to show that it is actually polynomial-time solvable for this graph class too. We also study the parameterized complexity of the problems and show some tractability and intractability results.

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Notes

  1. 1.

    We consider parameterized complexity in Sect. 5 where the equivalence does not hold.

  2. 2.

    The ordinary MSO is enough for our purpose. We introduce CMSO to precisely state Proposition 3.4.

  3. 3.

    The result is presented in a more general way in the original paper (see [2, Theorem 1]).

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Acknowledgements

The authors thank the anonymous reviewers for constructive comments that improved the presentation of the paper.

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Correspondence to Yota Otachi.

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Partially supported by NETWORKS project and by MEXT/JSPS KAKENHI Grant Numbers 24106004, 24220003, 25730003, 26540005. Yota Otachi was partially supported by FY 2015 Researcher Exchange Program between JSPS and NSERC.

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Bodlaender, H.L., Ono, H. & Otachi, Y. Degree-Constrained Orientation of Maximum Satisfaction: Graph Classes and Parameterized Complexity. Algorithmica 80, 2160–2180 (2018). https://doi.org/10.1007/s00453-017-0399-9

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Keywords

  • Orientation
  • Graph class
  • Width parameter
  • Parameterized complexity