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arXiv · 2605.03893

Optimal Hardness of Online Algorithms for Large Common Induced Subgraphs

David Gamarnik·Miklós Z. Rácz·Gabe Schoenbach
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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.

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David Gamarnik, Miklós Z. Rácz, Gabe Schoenbach. 2026-05-05. Optimal Hardness of Online Algorithms for Large Common Induced Subgraphs. https://arxiv.org/abs/2605.03893

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