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

作者：Bodlaender Hans L; Ono Hirotaka; Otachi Yota^*

来源：Algorithmica, 2018, 80(7): 2160-2180.

DOI：10.1007/s00453-017-0399-9

• Orientation
• Graph class
• Width parameter
• Parameterized complexity

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.

• 出版日期2018-7