Optimal Hardness of Online Algorithms for Large Common Induced Subgraphs

2026-05-05

AI Summary

This work investigates the computational limits of online algorithms for finding the largest common induced subgraph in two independent $G(n,1/2)$ Erdős–Rényi random graphs. By combining the overlap gap property (OGP) with an interpolation argument, the authors establish that no online algorithm can, with non-vanishing probability, recover a common induced subgraph larger than $(2+\varepsilon)\log_2 n$. Complementing this impossibility result, they propose a simple greedy online algorithm that achieves a subgraph of size $(2 - o(1))\log_2 n$, thereby demonstrating that this bound is optimal among all online strategies and matches one-half of the information-theoretic upper limit. These findings reveal a substantial gap between what is computationally achievable by online methods and the fundamental information-theoretic threshold for this problem.

Abstract

We study the problem of efficiently finding large common induced subgraphs of two independent Erdős--Rényi 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, Kizildağ, and Warnke that connects OGP and online algorithms.

Problem

• online algorithms
• common induced subgraphs
• random graphs
• computation-to-optimization gap
• overlap gap property

Innovation

• online algorithms
• overlap gap property
• computation-to-optimization gap
• random graphs
• induced subgraphs