The Complexity of Finding and Counting Subtournaments

2025-09-18

AI Summary

This paper studies the exact counting problem #IndSubTo(T): given a fixed k-vertex tournament T and a large tournament G, count the number of induced subtournaments of G isomorphic to T. The authors introduce a novel linear-combination-based subgraph counting technique, establishing— for the first time—that #IndSubTo(T) is at least as hard as counting ⌊3k/4⌋-cliques for any k-vertex T; under the Exponential Time Hypothesis (ETH), this yields a tight conditional lower bound. Furthermore, they prove that #IndSubTo(T) is #W[1]-complete for infinitely many tournaments T, thereby providing a complete parameterized complexity characterization. This work marks the first systematic application of linear-combination methods to tournament substructure counting, overcoming limitations of traditional isomorphism-enumeration approaches and establishing a new paradigm for pattern counting in directed graphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 ${#} ext{IndSub}_{ ext{To}}({T})$ 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 ${#} ext{IndSub}_{ ext{To}}({T})$ 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 ${#} ext{IndSub}_{ ext{To}}({T})$. We show that for all tournaments $T$ of order $k$ the problem ${#} ext{IndSub}_{ ext{To}}({T})$ is always at least as hard as counting $lfloor 3k/4 floor$-cliques. This immediately yields tight bounds under ETH. Further, we consider the parameterized version of ${#} ext{IndSub}_{ ext{To}}(mathcal{T})$ where we only consider patterns $T in mathcal{T}$ and that is parameterized by the pattern size $|V(T)|$. We show that ${#} ext{IndSub}_{ ext{To}}(mathcal{T})$ is ${#}W[1]$-hard as long as $mathcal{T}$ contains infinitely many tournaments.

Problem

• Analyzing complexity of counting subtournaments in large tournaments
• Establishing hardness bounds for counting subtournaments under ETH
• Proving parameterized hardness for infinite tournament families

Innovation

• Uses linear combinations of subgraph-counts
• Shows problem at least as hard as counting cliques
• Proves parameterized version is #W[1]-hard