# Almost-Linear Planted Cliques Elude the Metropolis Process

arXiv: https://arxiv.org/abs/2204.01911 · HTML: https://mlsanity.com/paper/2204.01911/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2022.

## Built on by (most cited first, 7 of 7)

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- [On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions](https://mlsanity.com/paper/2209.02001/) (2022)
- [Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials](https://mlsanity.com/paper/2211.05274/) (2022)
- [A short review on the maximum clique problem algorithms with classical, AI, and quantum methods](https://mlsanity.com/paper/2403.09742/) (2024)
- [On Approximate Computation of Critical Points](https://mlsanity.com/paper/2601.21917/) (2026)
- [Recovery thresholds for hidden weighted sparse graphs](https://mlsanity.com/paper/2606.14335/) (2026)

## Abstract (arXiv)

A seminal work of Jerrum (1992) showed that large cliques elude the Metropolis process. More specifically, Jerrum showed that the Metropolis algorithm cannot find a clique of size k=Theta(nα), α ∈ (0,1/2), which is planted in the ErdHos-R'enyi random graph G(n,1/2), in polynomial time. Information theoretically it is possible to find such planted cliques as soon as k ≥ (2+ε) log n. Since the work of Jerrum, the computational problem of finding a planted clique in G(n,1/2) was studied extensively and many polynomial time algorithms were shown to find the planted clique if it is of size k = Omega(√n), while no polynomial-time algorithm is known to work when k=o(√n). Notably, the first evidence of the problem's algorithmic hardness is commonly attributed to the result of Jerrum from 1992. In this paper we revisit the original Metropolis algorithm suggested by Jerrum. Interestingly, we find that the Metropolis algorithm actually fails to recover a planted clique of size k=Theta(nα) for any constant 0 ≤ α < 1. Moreover, we strengthen Jerrum's results in a number of other ways including: Like many results in the MCMC literature, the result of Jerrum shows that there exists a starting state (which may depend on the instance) for which the Metropolis algorithm fails. For a wide range of temperatures, we show that the algorithm fails when started at the most natural initial state, which is the empty clique. This answers an open problem stated in Jerrum (1992). We also show that the simulated tempering version of the Metropolis algorithm, a more sophisticated temperature-exchange variant of it, also fails at the same regime of parameters. Finally, our results confirm recent predictions by Gamarnik and Zadik (2019) and Angelini, Fachin, de Feo (2021).