Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial

TL;DR

This paper challenges previous beliefs by showing that counting certain small induced subgraphs called scorpions can be done efficiently, revealing new insights into the complexity of subgraph counting problems.

Contribution

It demonstrates that scorpion graphs can be counted in polynomial time, refuting the conjecture that all non-meager properties are hard to count.

Findings

Scorpion graphs can be counted in $O(n^4)$ time for all $k$.

A variant achieves arbitrary intermediate complexity assuming ETH.

Refutes the conjecture that all non-meager properties are $ ext{ extsterling}$W[1]-hard.

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.