# Parallel Tempering for the planted clique problem

arXiv: https://arxiv.org/abs/1802.05903 · HTML: https://mlsanity.com/paper/1802.05903/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Feb 2018.

## Built on by (most cited first, 5 of 5)

- [Modern graph neural networks do worse than classical greedy algorithms in solving combinatorial optimization problems like maximum independent set](https://mlsanity.com/paper/2206.13211/) (2022)
- [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)
- [Stochastic Gradient Descent-like relaxation is equivalent to Metropolis dynamics in discrete optimization and inference problems](https://mlsanity.com/paper/2309.05337/) (2023)
- [Planted Dense Subgraphs in Dense Random Graphs Can Be Recovered using Graph-based Machine Learning](https://mlsanity.com/paper/2201.01825/) (2022)
- [A short review on the maximum clique problem algorithms with classical, AI, and quantum methods](https://mlsanity.com/paper/2403.09742/) (2024)

## Abstract (arXiv)

The theoretical information threshold for the planted clique problem is 2log2(N), however no polynomial algorithm is known to recover a planted clique of size O(N1/2-ε), ε>0. In this paper we will apply a standard method for the analysis of disordered models, the Parallel-Tempering (PT) algorithm, to the clique problem, showing numerically that its time-scaling in the hard region is indeed polynomial for the analyzed sizes. We also apply PT to a different but connected model, the Sparse Planted Independent Set problem. In this situation thresholds should be sharper and finite size corrections should be less important. Also in this case PT shows a polynomial scaling in the hard region for the recovery.