The Complexity of Finding and Counting Subtournaments
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912591863873536 |
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| author | Döring, Simon Houdaigoui, Sarah Picasarri-Arrieta, Lucas Wellnitz, Philip |
| author_facet | Döring, Simon Houdaigoui, Sarah Picasarri-Arrieta, Lucas Wellnitz, Philip |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14807 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Complexity of Finding and Counting Subtournaments Döring, Simon Houdaigoui, Sarah Picasarri-Arrieta, Lucas Wellnitz, Philip Computational Complexity Data Structures and Algorithms 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. |
| title | The Complexity of Finding and Counting Subtournaments |
| topic | Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2509.14807 |