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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.

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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)