Abstract
A simple digraph is semicomplete if for any two of its vertices u and v , at least one of the arcs ( u , v ) and ( v , u ) is present. We study the complexity of computing two layout parameters of semicomplete digraphs: cutwidth and optimal linear arrangement (O la ). We prove the following: • Both parameters are NP-hard to compute and the known exact and parameterized algorithms for them have essentially optimal running times, assuming the Exponential Time Hypothesis. • The cutwidth parameter admits a quadratic Turing kernel, whereas it does not admit any polynomial kernel unless NP ⊆ coNP/poly. By contrast, O la admits a linear kernel. These results essentially complete the complexity analysis of computing cutwidth and O la on semicomplete digraphs (with respect to standard parameters). Our techniques also can be used to analyze the sizes of minimal obstructions for having a small cutwidth under the induced subdigraph relation.
Highlights
A directed graph is simple if it does not contain a self-loop or multiple arcs with the same head and tail
Tournaments and semi-complete digraphs form a rich and interesting subclass of directed graphs; we refer to the book of Bang-Jensen and Gutin [1] for an overview
We study two layout parameters for tournaments and semi-complete digraphs: cutwidth and optimal linear arrangement (Ola)
Summary
A directed graph (digraph) is simple if it does not contain a self-loop or multiple arcs with the same head and tail. Unless NP ⊆ coNP/poly, there exists no polynomial-size kernelization algorithm for the problem of computing the cutwidth of a semi-complete digraph. The proofs of these two theorems directly follow from the understanding of the contribution of strongly connected components in optimal orderings. We prove that the problem of computing the cutwidth of a semi-complete digraph admits a quadratic Turing kernel, which is encapsulated in the following theorem. There exists a polynomial-time algorithm that given a semi-complete digraph D, outputs an ordering of its vertices of width, and respectively cost, upper bounded by twice the cutwidth, respectively Ola-cost, of D. The proofs of statements marked with ♠ will appear in the full version of the paper
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