On optimal distinguishers for Planted Clique

In a distinguishing problem, the input is a sample drawn from one of two distributions and the algorithm is tasked with identifying the source distribution. The performance of a distinguishing algorithm is measured by its advantage, i.e., its incremental probability of success over a random guess. A classic example of a distinguishing problem is the Planted Clique problem, where the input is a graph sampled from either $G(n,1/2)$ -- the standard Erd\H{o}s-R\'{e}nyi model, or $G(n,1/2,k)$ -- the Erd\H{o}s-R\'{e}nyi model with a clique planted on a random subset of $k$ vertices. The Planted Clique Hypothesis asserts that efficient algorithms cannot achieve advantage better than some absolute constant, say $1/4$, whenever $k=n^{1/2-\Omega(1)}$. In this work, we aim to precisely understand the optimal distinguishing advantage achievable by efficient algorithms on Planted Clique. We show the following results under the Planted Clique hypothesis: 1. Optimality of low-degree polynomials: No efficient algorithm can beat the advantage the optimal low-degree polynomial. Concretely, this means that the advantage of any efficient algorithm is at most $(1+o(1))\cdot k^2/(\sqrt{\pi}n)$, which is optimal in light of a simple edge-counting algorithm achieving this bound. 2. Harder planted distributions: There is an efficiently sampleable distribution $\mathcal{P}^*$ supported on graphs containing $k$-cliques such that no efficient algorithm can distinguish $\mathcal{P}^*$ from $G(n,1/2)$ with advantage $n^{-d}$ for an arbitrarily large constant $d$. In other words, there exist alternate planted distributions that are much harder than $G(n,1/2,k)$. Along the way, we prove a constructive hard-core lemma for a broad class of distributions with respect to low-degree polynomials. This result is applicable much more widely beyond Planted Clique and might be of independent interest.