Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs

08/27/2020
by   Afonso S. Bandeira, et al.
0

We study the problem of efficiently refuting the k-colorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, showing optimality of a simple spectral certificate. This evidence takes the form of a computationally-quiet planting: we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishable by a large class of algorithms. We generalize our results to the more general problem of certifying an upper bound on the maximum k-cut. This quiet planting is achieved by minimizing the effect of the planted structure (e.g. colorings or cuts) on the graph spectrum. Specifically, the planted structure corresponds exactly to eigenvectors of the adjacency matrix. This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. To illustrate this further, we give similar results for a Gaussian analogue of this problem: a quiet version of the spiked model, where we plant an eigenspace rather than adding a generic low-rank perturbation. Our evidence for computational hardness of distinguishing two distributions is based on three different heuristics: stability of belief propagation, the local statistics hierarchy, and the low-degree likelihood ratio. Of independent interest, our results include general-purpose bounds on the low-degree likelihood ratio for multi-spiked matrix models, and an improved low-degree analysis of the stochastic block model.

READ FULL TEXT

page 1

page 2

page 3

page 4

02/19/2019

Computational Hardness of Certifying Bounds on Constrained PCA Problems

Given a random n × n symmetric matrix W drawn from the Gaussian orthogo...
11/05/2019

Local Statistics, Semidefinite Programming, and Community Detection

We propose a new hierarchy of semidefinite programming relaxations for i...
01/22/2018

Even flying cops should think ahead

We study the entanglement game, which is a version of cops and robbers, ...
11/13/2019

On the Relativized Alon Second Eigenvalue Conjecture VI: Sharp Bounds for Ramanujan Base Graphs

This is the sixth in a series of articles devoted to showing that a typi...
07/26/2019

Notes on Computational Hardness of Hypothesis Testing: Predictions using the Low-Degree Likelihood Ratio

These notes survey and explore an emerging method, which we call the low...
07/04/2019

Vector Colorings of Random, Ramanujan, and Large-Girth Irregular Graphs

We prove that in sparse Erdős-Rényi graphs of average degree d, the vect...
01/19/2018

Joint estimation of parameters in Ising model

We study joint estimation of the inverse temperature and magnetization p...