Sum-of-Squares Lower Bounds for Independent Set on Ultra-Sparse Random Graphs

Pravesh K. Kothari, Aaron Potechin, Jeff Xu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We prove that for every D ∈ N, and large enough constant d ∈ N, with high probability over the choice of G ∼G(n,d/n), the Erdos-Renyi random graph distribution, the canonical degree 2D Sum-of-Squares relaxation fails to certify that the largest independent set in G is of size o(n/√d D4). In particular, degree D sum-of-squares strengthening can reduce the integrality gap of the classical theta SDP relaxation by at most a O(D4) factor. This is the first lower bound for >4-degree Sum-of-Squares (SoS) relaxation for any problems on ultra sparse random graphs (i.e. average degree of an absolute constant). Such ultra-sparse graphs were a known barrier for previous methods and explicitly identified as a major open direction. Indeed, the only other example of an SoS lower bound on ultra-sparse random graphs was a degree-4 lower bound for Max-Cut. Our main technical result is a new method to obtain spectral norm estimates on graph matrices (a class of low-degree matrix-valued polynomials in G(n,d/n)) that are accurate to within an absolute constant factor. All prior works lose log n factors that trivialize any lower bound on o(logn)-degree random graphs. We combine these new bounds with several upgrades on the machinery for analyzing lower-bound witnesses constructed by pseudo-calibration so that our analysis does not lose any ω(1)-factors that would trivialize our results. In addition to other SoS lower bounds, we believe that our methods for establishing spectral norm estimates on graph matrices will be useful in the analyses of numerical algorithms on average-case inputs.

Original languageEnglish (US)
Title of host publicationSTOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing
EditorsBojan Mohar, Igor Shinkar, Ryan O�Donnell
PublisherAssociation for Computing Machinery
Pages1923-1934
Number of pages12
ISBN (Electronic)9798400703836
DOIs
StatePublished - Jun 10 2024
Externally publishedYes
Event56th Annual ACM Symposium on Theory of Computing, STOC 2024 - Vancouver, Canada
Duration: Jun 24 2024Jun 28 2024

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference56th Annual ACM Symposium on Theory of Computing, STOC 2024
Country/TerritoryCanada
CityVancouver
Period6/24/246/28/24

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Average-case complexity
  • Random Matrix Theory
  • Sum-of-Squares Lower Bounds

Fingerprint

Dive into the research topics of 'Sum-of-Squares Lower Bounds for Independent Set on Ultra-Sparse Random Graphs'. Together they form a unique fingerprint.

Cite this