The Complexity of Finding and Counting Subtournaments

Abstract

We study the complexity of counting and finding small tournament patterns inside large tournaments. Given a fixed tournament $T$ of order $k$, we write $\#{\text{IndSub}}_{\text{To}}(\{T\})$ for the problem whose input is a tournament $G$ and the task is to compute the number of subtournaments of $G$ that are isomorphic to $T$. Previously, Yuster [Yus25] obtained that $\#{\text{IndSub}}_{\text{To}}(\{T\})$ is hard to compute for random tournaments $T$. We consider a new approach that uses linear combinations of subgraph-counts [CDM17] to obtain a finer analysis of the complexity of $\#{\text{IndSub}}_{\text{To}}(\{T\})$. We show that for all tournaments $T$ of order $k$ the problem $\#{\text{IndSub}}_{\text{To}}(\{T\})$ is always at least as hard as counting $\lfloor 3k/4\rfloor$-cliques. This immediately yields tight bounds under ETH. Further, we consider the parameterized version of $\#{\text{IndSub}}_{\text{To}}(\mathcal{T})$ where we only consider patterns $T\in \mathcal{T}$ and that is parameterized by the pattern size $|V(T)|$. We show that $\#{\text{IndSub}}_{\text{To}}(\mathcal{T})$ is $\#W[1]$-hard as long as $\mathcal{T}$ contains infinitely many tournaments.