Sum-of-Squares Lower Bounds for Sparse Independent Set

11/17/2021
by   Chris Jones, et al.
0

The Sum-of-Squares (SoS) hierarchy of semidefinite programs is a powerful algorithmic paradigm which captures state-of-the-art algorithmic guarantees for a wide array of problems. In the average case setting, SoS lower bounds provide strong evidence of algorithmic hardness or information-computation gaps. Prior to this work, SoS lower bounds have been obtained for problems in the "dense" input regime, where the input is a collection of independent Rademacher or Gaussian random variables, while the sparse regime has remained out of reach. We make the first progress in this direction by obtaining strong SoS lower bounds for the problem of Independent Set on sparse random graphs. We prove that with high probability over an Erdos-Renyi random graph G∼ G_n,d/n with average degree d>log^2 n, degree-D_SoS SoS fails to refute the existence of an independent set of size k = Ω(n/√(d)(log n)(D_SoS)^c_0) in G (where c_0 is an absolute constant), whereas the true size of the largest independent set in G is O(nlog d/d). Our proof involves several significant extensions of the techniques used for proving SoS lower bounds in the dense setting. Previous lower bounds are based on the pseudo-calibration heuristic of Barak et al [FOCS 2016] which produces a candidate SoS solution using a planted distribution indistinguishable from the input distribution via low-degree tests. In the sparse case the natural planted distribution does admit low-degree distinguishers, and we show how to adapt the pseudo-calibration heuristic to overcome this. Another notorious technical challenge for the sparse regime is the quest for matrix norm bounds. In this paper, we obtain new norm bounds for graph matrices in the sparse setting.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

11/09/2020

Machinery for Proving Sum-of-Squares Lower Bounds on Certification Problems

In this paper, we construct general machinery for proving Sum-of-Squares...
12/02/2019

Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

We show exponential lower bounds on resolution proof length for pigeonho...
09/15/2020

Positivity-preserving extensions of sum-of-squares pseudomoments over the hypercube

We introduce a new method for building higher-degree sum-of-squares lowe...
08/20/2021

Signal Detection in Degree Corrected ERGMs

In this paper, we study sparse signal detection problems in Degree Corre...
08/13/2013

Community Detection in Sparse Random Networks

We consider the problem of detecting a tight community in a sparse rando...
11/12/2020

Sparse PCA: Algorithms, Adversarial Perturbations and Certificates

We study efficient algorithms for Sparse PCA in standard statistical mod...
09/04/2018

SOS lower bounds with hard constraints: think global, act local

Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficie...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.