Exploring the Complexity of Layout Parameters in Tournaments and Semicomplete Digraphs

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.