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

## TL;DR

This paper investigates the computational complexity of two layout parameters, cutwidth and optimal linear arrangement, in semi-complete digraphs, establishing NP-hardness and kernelization bounds, thus completing their complexity analysis.

## Contribution

It proves NP-hardness for both parameters, analyzes kernelization limits, and provides a linear kernel for OLA, advancing understanding of these problems in semi-complete digraphs.

## Key findings

- Both parameters are NP-hard to compute.
- Cutwidth admits a quadratic Turing kernel but no polynomial kernel.
- OLA admits a linear kernel.

## Abstract

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.