Exploring the complexity of layout parameters in tournaments and semi-complete digraphs

A simple digraph is semi-complete 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 semi-complete digraphs: cutwidth and optimal linear arrangement (OLA). We prove that: (1) Both parameters are $\mathsf{NP}$-hard to compute and the known exact and parameterized algorithms for them have essentially optimal running times, assuming the Exponential Time Hypothesis; (2) The cutwidth parameter admits a quadratic Turing kernel, whereas it does not admit any polynomial kernel unless $\mathsf{NP}\subseteq \mathsf{coNP}/\textrm{poly}$. By contrast, OLA admits a linear kernel. These results essentially complete the complexity analysis of computing cutwidth and OLA on semi-complete digraphs. Our techniques can be also used to analyze the sizes of minimal obstructions for having small cutwidth under the induced subdigraph relation.