DeepAI AI Chat
Log In Sign Up

The Landscape of the Planted Clique Problem: Dense subgraphs and the Overlap Gap Property

04/15/2019
by   David Gamarnik, et al.
MIT
0

In this paper we study the computational-statistical gap of the planted clique problem, where a clique of size k is planted in an Erdos Renyi graph G(n,1/2) resulting in a graph G(n,1/2,k). The goal is to recover the planted clique vertices by observing G(n,1/2,k) . It is known that the clique can be recovered as long as k ≥(2+ϵ) n for any ϵ>0, but no polynomial-time algorithm is known for this task unless k=Ω(√(n)). Following a statistical-physics inspired point of view as an attempt to understand this computational-statistical gap, we study the landscape of the "sufficiently dense" subgraphs of G and their overlap with the planted clique. Using the first moment method, we study the densest subgraph problems for subgraphs with fixed, but arbitrary, overlap size with the planted clique, and provide evidence of a phase transition for the presence of Overlap Gap Property (OGP) at k=Θ(√(n)). OGP is a concept introduced originally in spin glass theory and known to suggest algorithmic hardness when it appears. We establish the presence of OGP when k is a small positive power of n by using a conditional second moment method. As our main technical tool, we establish the first, to the best of our knowledge, concentration results for the K-densest subgraph problem for the Erdos-Renyi model G(n,1/2) when K=n^0.5-ϵ for arbitrary ϵ>0. Finally, to study the OGP we employ a certain form of overparametrization, which is conceptually aligned with a large body of recent work in learning theory and optimization.

READ FULL TEXT

page 1

page 2

page 3

page 4

12/07/2022

Densest Subgraphs of a Dense Erdös-Rényi Graph. Asymptotics, Landscape and Universality

We consider the problem of estimating the edge density of densest K-node...
03/02/2021

Algorithmic Obstructions in the Random Number Partitioning Problem

We consider the algorithmic problem of finding a near-optimal solution f...
11/17/2020

Exact recovery of Planted Cliques in Semi-random graphs

In this paper, we study the Planted Clique problem in a semi-random mode...
08/26/2019

The Overlap Gap Property in Principal Submatrix Recovery

We study support recovery for a k × k principal submatrix with elevated ...
07/10/2020

Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

A graph G is weakly γ-closed if every induced subgraph of G contains one...
04/05/2022

Almost-Linear Planted Cliques Elude the Metropolis Process

A seminal work of Jerrum (1992) showed that large cliques elude the Metr...
08/28/2020

Is the space complexity of planted clique recovery the same as that of detection?

We study the planted clique problem in which a clique of size k is plant...