Improved polynomial-time algorithms for detecting and recovering planted $Θ(\sqrt{n})$-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 $k = Θ(\sqrt{n})$ 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 $t \geq 1$, 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 $k = λ\sqrt{n}$ planted in a random location with $λ&gt; 1 / \sqrt{c(t)}$ 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 $\widetilde{O}(n<sup>2)$-time message-passing algorithm of Deshpande--Montanari (2015) that succeeds when $λ&gt; 1 / \sqrt{e} \approx 0.6066$. For $t \geq 3$, 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 $λ$. We further show that the above algorithm for $t = 2$ can be implemented in time $n^{ω+ o(1)}$ for $ω$ the constant of square matrix multiplication and succeeds when $λ&gt; 0.3320$; under the folklore conjecture that $ω= 2$, 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 $λ$. Taken together, our results achieve the best known tradeoff between runtime and signal strength $λ$.