Forbidden Tournaments and the Orientation Completion Problem

TL;DR

This paper classifies the computational complexity of determining whether a graph can be oriented to avoid certain fixed tournaments, showing it is either polynomial-time solvable or NP-complete.

Contribution

It provides a complete complexity classification for the F-free orientation problem for any finite set of tournaments, linking it to the orientation completion problem.

Findings

The problem is in P or NP-complete for any fixed set of tournaments.

Reduces classification to the orientation completion problem.

Uses tools from constraint satisfaction and permutation group theory.

Abstract

For a fixed finite set of finite tournaments ${\mathcal F}$, the ${\mathcal F}$-free orientation problem asks whether a given finite undirected graph $G$ has an $\mathcal F$-free orientation, i.e., whether the edges of $G$ can be oriented so that the resulting digraph does not embed any of the tournaments from ${\mathcal F}$. We prove that for every ${\mathcal F}$, this problem is in P or NP-complete. Our proof reduces the classification task to a complete complexity classification of the orientation completion problem for ${\mathcal F}$, which is the variant of the problem above where the input is a directed graph instead of an undirected graph, introduced by Bang-Jensen, Huang, and Zhu (2017). Our proof uses results from the theory of constraint satisfaction, and a result of Agarwal and Kompatscher (2018) about infinite permutation groups and transformation monoids.