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.

翻译：我们研究了高效寻找两个独立 Erdős--Rényi 随机图 $G_1, G_2 \sim \mathbb{G}(n,1/2)$ 的大规模公共诱导子图的问题。近期，Chatterjee 和 Diaconis 证明，$G_1$ 和 $G_2$ 的最大公共诱导子图以高概率具有 $(4-o(1))\log_2 n$ 的大小。我们首先证明，一个简单的贪婪在线算法能够以高概率找到 $G_1$ 和 $G_2$ 的一个规模为 $(2-o(1)) \log_2 n$ 的公共诱导子图。我们的主要结果表明，当 $n \to \infty$ 时，没有任何在线算法能够以远离 $0$ 的概率找到 $G_1$ 和 $G_2$ 的规模至少为 $(2+\varepsilon) \log_2 n$ 的公共诱导子图。这些结果共同表明，该问题存在计算与优化之间的差距。为证明这一不可能性结果，我们展示了该问题的解空间具有（多重）重叠间隙性质（OGP）的一个版本，并利用了 Gamarnik、Kizildağ 和 Warnke 最新发展的将 OGP 与在线算法联系起来的插值论证方法。