Efficient algorithms for planted clique below the √n threshold

Develop a polynomial-time algorithm that detects and recovers the planted clique in an Erdős–Rényi graph G(n, 1/2) when the clique size k satisfies (2+ε) log n ≤ k = o(√n), thereby closing the statistical-to-computational gap between information-theoretic recovery (via brute force) and currently known efficient methods that require k ≳ √n.

Background

The paper introduces average-case complexity through the planted clique problem in Erdős–Rényi graphs and highlights the well-known statistical-to-computational gap: brute-force methods can recover a planted clique of size above approximately (2+ε) log n, while all known efficient (polynomial-time) algorithms require the clique size to be on the order of √n or larger.

This gap is a central open challenge in average-case complexity and statistical inference, with broad implications for understanding algorithmic thresholds relative to information-theoretic limits.

References

While it is possible to detect and recover the hidden clique by brute-force search when its size exceeds $(2+\varepsilon) \log n$, where $n$ is the number of vertices and $\varepsilon$ is an arbitrary positive constant, we are unaware of any efficient algorithm to do so unless the clique size exceeds $\Omega(\sqrt{n})$, see \Cref{fig:clique}.

— Average-case complexity in statistical inference: A puzzle-driven research seminar  (2506.22182 - Kireeva et al., 27 Jun 2025) in Introduction (Figure 2 reference)

We therefore believe the following is a fundamental question about the planted clique problem. Determine $\tau_{\mathrm{rand}(\lambda)$, $\tau_{\det,\mathrm{avg}(\lambda)$, and $\tau_{\det}(\lambda)$ for every $\lambda>0$.

— Improved polynomial-time algorithms for detecting and recovering planted $Θ(\sqrt{n})$-cliques  (2609.24780 - Kunisky et al., 21 Sep 2026) in Question 1, Section 1, subsection “Fine-grained polynomial-time complexity”

For every fixed $\varepsilon\in(0,1/2)$, no polynomial-time algorithm achieves strong detection, and hence no polynomial-time algorithm achieves strong recovery, in the planted clique problem when $k\le n{1/2-\varepsilon}$.

— Improved polynomial-time algorithms for detecting and recovering planted $Θ(\sqrt{n})$-cliques  (2609.24780 - Kunisky et al., 21 Sep 2026) in Conjecture 1, Section 1, subsection “Main results”

...it is natural to ask whether the threshold $\sqrt{n / e}$ has a fundamental computational meaning or is instead only relevant for our specific algorithm.

— Improved polynomial-time algorithms for detecting and recovering planted $Θ(\sqrt{n})$-cliques  (2609.24780 - Kunisky et al., 21 Sep 2026) in Section 1, subsection “Summary of consequences of prior results,” paragraph “Message passing and nearly-linear time algorithms”