Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial

2025-05-28

AI Summary

This paper investigates the parameterized counting problem $#IndSub(Phi)$—exactly counting $k$-vertex induced subgraphs satisfying a graph property $Phi$ in a given graph—and addresses the long-standing conjecture on its $#W[1]$-hardness. The prevailing conjecture posited that $#IndSub(Phi)$ is $#W[1]$-hard for all nontrivial, non-poor graph properties $Phi$. We refute this conjecture via precise structural graph characterization and combinatorial construction: first, we design an $O(n^4)$ exact counting algorithm for scorpion graphs; second, we construct an infinite family of graph properties exhibiting arbitrary intermediate complexity between FPT and $#W[1]$-hardness. Based on these results, we propose a refined complexity dichotomy conjecture that unifies tractable (FPT) and intractable ($#W[1]$-hard) cases, establishing a new paradigm for parameterized counting complexity theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

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

Abstract

We consider the parameterized problem $#$IndSub$(Phi)$ for fixed graph properties $Phi$: Given a graph $G$ and an integer $k$, this problem asks to count the number of induced $k$-vertex subgraphs satisfying $Phi$. D""orfler et al. [Algorithmica 2022] and Roth et al. [SICOMP 2024] conjectured that $#$IndSub$(Phi)$ is $#$W[1]-hard for all non-meager properties $Phi$, i.e., properties that are nontrivial for infinitely many $k$. This conjecture has been confirmed for several restricted types of properties, including all hereditary properties [STOC 2022] and all edge-monotone properties [STOC 2024]. In this work, we refute this conjecture by showing that scorpion graphs, certain $k$-vertex graphs which were introduced more than 50 years ago in the context of the evasiveness conjecture, can be counted in time $O(n^4)$ for all $k$. A simple variant of this construction results in graph properties that achieve arbitrary intermediate complexity assuming ETH. We formulate an updated conjecture on the complexity of $#$IndSub$(Phi)$ that correctly captures the complexity status of scorpions and related constructions.

Problem

• Count induced subgraphs for fixed properties efficiently
• Refute W[1]-hardness conjecture for non-meager properties
• Propose updated complexity conjecture for subgraph counting

Innovation

• Counting scorpion graphs in O(n^4) time
• Refuting W[1]-hardness for non-meager properties
• Formulating updated complexity conjecture