Improved polynomial-time algorithms for detecting and recovering planted $\Theta (\sqrt{n})$-cliques

Abstract

In the planted clique problem, one observes either an Erdős--Rényi graph on $n$ vertices or such a graph with a clique added to $k=k(n)$ vertices, and seeks to detect or recover the clique. It is widely believed that $k=\Theta (\sqrt{n})$ is the smallest clique size for which polynomial-time algorithms exist for these tasks. We develop new algorithms in this regime using color-coding to estimate signed subgraph counts, further accelerated with fast matrix multiplication. We first show that, for each $t\ge 1$, for $c(t)$ a constant associated to the order of growth of the number of connected graphs of treewidth at most $t$, cliques of size $k=\lambda \sqrt{n}$ planted in a random location with $\lambda >1/\sqrt{c(t)}$ can be detected and recovered in time $n^{t+1+o(1)}$. For instance, since $c(1)=e$, this recovers by counting signed trees the performance of the $\tilde{O}(n^2)$-time message-passing algorithm of Deshpande--Montanari (2015) that succeeds when $\lambda >1/\sqrt{e}\approx 0.6066$. For $t\ge 3$, the exact value of $c(t)$ is not known, but lower bounds on it give a hierarchy of slower polynomial-time algorithms that succeed for smaller $\lambda$. We further show that the above algorithm for $t=2$ can be implemented in time $n^{\omega +o(1)}$ for $\omega$ the constant of square matrix multiplication and succeeds when $\lambda >0.3320$; under the folklore conjecture that $\omega =2$, this runs in the nearly-linear time of the algorithm of Deshpande--Montanari while finding smaller cliques. Second, we show that the above algorithm for $t=1$ can be combined with the boosting scheme of Alon--Krivelevich--Sudakov (1998) using rectangular matrix multiplication, giving improved runtimes for smaller $\lambda$. Taken together, our results achieve the best known tradeoff between runtime and signal strength $\lambda$.

77 pages