Almost-Linear Planted Cliques Elude the Metropolis Process

TL;DR

This paper demonstrates that the Metropolis algorithm, including its tempering variant, fails to find planted cliques of size smaller than linear in the number of nodes in Erdős-Rényi graphs, confirming longstanding conjectures about its limitations.

Contribution

The paper revisits Jerrum's original Metropolis algorithm and proves it fails for all sublinear planted clique sizes, also confirming the failure of simulated tempering in this regime.

Findings

Metropolis algorithm fails for clique size $k=\Theta(n^{\alpha})$, $0\leq \alpha<1$

Starting from the empty clique, the algorithm does not recover planted cliques in the sublinear regime

Simulated tempering also fails to find small planted cliques

Abstract

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 $k=\Theta(n^{\alpha}), \alpha \in (0,1/2)$, which is planted in the Erd\H{o}s-R\'{e}nyi random graph $G(n,1/2)$, in polynomial time. Information theoretically it is possible to find such planted cliques as soon as $k \ge (2+\epsilon) \log n$. 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 $k = \Omega(\sqrt{n})$, while no polynomial-time algorithm is known to work when $k=o(\sqrt{n})$. 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 $k=\Theta(n^{\alpha})$ for any constant $0 \leq \alpha < 1$. 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).