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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The authors thank the anonymous reviewers for constructive comments that improved the presentation of the paper.
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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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DOI: https://doi.org/10.1007/s00453-017-0399-9