Optimal Hardness of Online Algorithms for Large Common Induced Subgraphs

TL;DR

This paper investigates the limits of online algorithms in finding large common induced subgraphs in random graphs, revealing a fundamental gap between what simple algorithms can achieve and the theoretical maximum.

Contribution

It introduces a new lower bound for online algorithms and demonstrates a computation-to-optimization gap using the overlap gap property framework.

Findings

A greedy online algorithm finds subgraphs of size approximately 2 log n.

No online algorithm can reliably find subgraphs larger than (2+ε) log n.

The problem exhibits a computation-to-optimization gap supported by the overlap gap property.

Abstract

We study the problem of efficiently finding large common induced subgraphs of two independent Erd\H{o}s--R\'enyi random graphs $G_1, G_2 \sim \mathbb{G}(n,1/2)$. Recently, Chatterjee and Diaconis showed that the largest common induced subgraph of $G_1$ and $G_2$ has size $(4-o(1))\log_2 n$ with high probability. We first show that a simple greedy online algorithm finds a common induced subgraph of $G_1$ and $G_2$ of size $(2-o(1)) \log_2 n$ with high probability. Our main result shows that no online algorithm can find a common induced subgraph of $G_1$ and $G_2$ of size at least $(2+\varepsilon) \log_2 n$ with probability bounded away from $0$ as $n \to \infty$. Together, these results provide evidence that this problem exhibits a computation-to-optimization gap. To prove the impossibility result, we show that the solution space of the problem exhibits a version of the (multi) overlap gap property (OGP), and utilize an interpolation argument recently developed by Gamarnik, Kizilda\u{g}, and Warnke that connects OGP and online algorithms.