License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2021.26
URN: urn:nbn:de:0030-drops-143000
Go to the corresponding LIPIcs Volume Portal

Pang, Shuo

SOS Lower Bound for Exact Planted Clique

LIPIcs-CCC-2021-26.pdf (1 MB)


We prove a SOS degree lower bound for the planted clique problem on the Erdös-Rényi random graph G(n,1/2). The bound we get is degree d = Ω(ε²log n/log log n) for clique size ω = n^{1/2-ε}, which is almost tight. This improves the result of [Barak et al., 2019] for the "soft" version of the problem, where the family of the equality-axioms generated by x₁+...+x_n = ω is relaxed to one inequality x₁+...+x_n ≥ ω.
As a technical by-product, we also "naturalize" certain techniques that were developed and used for the relaxed problem. This includes a new way to define the pseudo-expectation, and a more robust method to solve out the coarse diagonalization of the moment matrix.

BibTeX - Entry

  author =	{Pang, Shuo},
  title =	{{SOS Lower Bound for Exact Planted Clique}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{26:1--26:63},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-143000},
  doi =		{10.4230/LIPIcs.CCC.2021.26},
  annote =	{Keywords: Sum-of-Squares, planted clique, random graphs, average-case lower bound}

Keywords: Sum-of-Squares, planted clique, random graphs, average-case lower bound
Collection: 36th Computational Complexity Conference (CCC 2021)
Issue Date: 2021
Date of publication: 08.07.2021

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI