Almost-Linear Planted Cliques Elude the Metropolis Process

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)) for alpha is an element of (0, 1/2), which is planted in the Erdos-Renyi G(n, 1/2), in polynomial-time. Information theoretically, it is possible to find such planted cliques when k >= (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(root n), while no polynomial-time algorithm is known to work when k = o(root n). The planted clique problem for k = o(root n) is now widely considered a foundational problem in the study of computational-statistical gaps. Notably, the first evidence of the problem's algorithmic hardness is commonly attributed to Jerrum (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 alpha is an element of (0, 1), unlike many other efficient algorithms that succeed when alpha > 1/2. Moreover, like many results in the MCMC literature, the result of Jerrum shows that there exists a starting state 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 from Jerrum (1992).