Induced Cycles and Paths Are Harder Than You Think

TL;DR

This paper establishes new complexity lower bounds for the Subgraph Isomorphism problem with fixed patterns, showing it is harder than previously known, especially for paths, cycles, and induced subgraphs, based on the $k$-Clique hypothesis.

Contribution

It provides refined hardness results for induced subgraph detection, improving previous bounds and removing reliance on the Hadwiger conjecture for certain cases.

Findings

SI for core patterns is as hard as $t$-Clique, where $t$ is the largest clique minor size.

Induced $k$-Path and $k$-Cycle detection are harder than previously established, requiring larger clique sizes.

Detecting induced 4-cycles requires near-quadratic time, even in sparse graphs.

Abstract

The goal of the paper is to give fine-grained hardness results for the Subgraph Isomorphism (SI) problem for fixed size induced patterns $H$, based on the $k$-Clique hypothesis that the current best algorithms for Clique are optimal. Our first main result is that for any pattern graph $H$ that is a {\em core}, the SI problem for $H$ is at least as hard as $t$-Clique, where $t$ is the size of the largest clique minor of $H$. This improves (for cores) the previous known results [Dalirrooyfard-Vassilevska W. STOC'20] that the SI for $H$ is at least as hard as $k$-clique where $k$ is the size of the largest clique {\em subgraph} in $H$, or the chromatic number of $H$ (under the Hadwiger conjecture). For detecting \emph{any} graph pattern $H$, we further remove the dependency of the result of [Dalirrooyfard-Vassilevska W. STOC'20] on the Hadwiger conjecture at the cost of a sub-polynomial decrease in the lower bound. The result for cores allows us to prove that the SI problem for induced $k$-Path and $k$-Cycle is harder than previously known. Previously [Floderus et al. Theor. CS 2015] had shown that $k$-Path and $k$-Cycle are at least as hard to detect as a $\lfloor k/2\rfloor$-Clique. We show that they are in fact at least as hard as $3k/4-O(1)$-Clique, improving the conditional lower bound exponent by a factor of $3/2$. Finally, we provide a new conditional lower bound for detecting induced $4$-cycles: $n^{2-o(1)}$ time is necessary even in graphs with $n$ nodes and $O(n^{1.5})$ edges.