32 problems

Consider the planted clique problem in an Erdős–Rényi random graph on $n$ vertices, where the planted clique has size $k$. Planted clique conjecture. Recovery is computationally in…

The Planted Clique problem asks whether an Erdős–Rényi graph on $n$ vertices contains a planted clique of size $k$. Planted Clique conjecture. There is no polynomial-time algorithm…

the $k$-planted clique conjecture. If $G$ is an instance of $k\textsc{-PC}_D(n,k_n,p)$, then

Let $d\geq 2$ be a fixed integer. For a $d$-uniform hypergraph $G$, let $H_0$ denote the null distribution $G\sim\mathcal{G}_d(N,1/2)$ and let $H_1$ denote the planted-clique distr…

Hypergraph planted clique conjecture. Suppose that $p=1/2$ and that $D\geq 2$ is a fixed integer. If

Planted clique hypothesis. The ${\sf PC}$ problem cannot be solved in polynomial time for $k=o(\sqrt{n})$ when $\gamma=\frac12$.

Luo et al.'s exact recovery conjecture. This $\sqrt n$-scale boundary is optimal for polynomial-time exact recovery. The conjecture concerns the computational threshold separating…

Let $m$ and $k$ be positive integers with $k\in\{1,\ldots,m\}$. Construct an Erdős–Rényi graph on $m$ vertices by connecting each pair independently with probability $1/2$, then pl…

Let $G$ be an Erdős–Rényi graph with a planted clique of size $k$, and suppose the goal is to identify the planted clique from the resulting graph. Planted clique hardness conjectu…

Planted clique conjecture. No polynomial-time algorithm can find the planted clique.

The Gaussian Planted Submatrix problem has signal parameter $\beta$, while the Planted Clique problem is obtained from a centered Erdős–Rényi adjacency matrix with an inserted cliq…

Agnostic lower-bound conjecture. No agnostic algorithm can match the oracle case when

Let $p\in(0,1)$ be constant. For each $n$, let $A_n:G_n\to\{0,1\}$ be a randomized polynomial-time algorithm, and let $k_n$ be a sequence of positive integers satisfying … For the…

Let $G$simmathcal{G}m(d,1/2)$under the null hypothesis and$G$sim\mathcal{G}_m(d,1/2,\tau)$ under the alternative, where $m\geq 2$ is fixed, and let … be the sum of Type-I and Typ…

Let $H$ be an order-$K$ hypergraph in the HPC detection problem, with null distribution $H_0:H\sim\mathcal{H}_K(p,1/2)$ and planted-clique alternative…

Let ${\sf HG}^s(n,1/2)$ be the Erdős–Rényi distribution on $s$-uniform hypergraphs and ${\sf HPC}_{\sf D}^s(n,k)$ the hypothesis-testing problem obtained by using the unconditioned…

Let ${\sf HG}^s(n,1/2)$ be the Erdős–Rényi distribution on $s$-uniform hypergraphs and ${\sf clHG}^s(n,1/2,k)$ the planted clique distribution conditioned on the first $s-2$ vertic…

Let ${\sf clkG}(\ell k,1/2,k)$ be the $k$-partite planted clique distribution conditioned on vertex $1$ belonging to the planted clique, and let ${\sf clkPC}_{\sf D}(\ell k,k)$ be…

Let $G$sim${\sf G}(n,1/2)$ denote an Erdős–Rényi graph and ${\sf G}(n,1/2,k)$ the graph distribution with a uniformly random planted clique of size $k$. A randomized logspace algor…

Let $G$ be a random graph in the planted clique detection problem, with $N$ vertices and planted clique size $\kappa$. A polynomial-time test is an algorithm whose running time is…

Planted clique conjecture. When $\kappa=o(\sqrt{d})$, no polynomial-time algorithm can correctly distinguish between $H_0$ and $H_1$ with error probability strictly below…

Let $\mathcal{G}_d(N,1/2,\kappa)$ be the hypergraphic planted clique model on $N$ vertices with fixed integer $d\geq2$, and let $\mathcal{E}_{{\rm HPC}_D}$ denote the sum of Type-I…

Let $G$ be an Erdős–Rényi graph with a planted clique of size $\kappa$ among $N$ vertices, and let a test be an algorithm that distinguishes the planted and null graph distribution…

Let $\rho$ range over the distributions specified in the paper’s Specific Hardness Assumptions. Symmetry Conjecture. These distributions are symmetric enough for the stronger Secre…

Let $\rho$ be sufficiently symmetric, and let $p_\rho(s)$ denote the intersection-size distribution for two independent sets sampled from $\rho$. Stronger Secret Leakage Planted Cl…