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.2023.30
URN: urn:nbn:de:0030-drops-183008
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Kunisky, Dmitriy ; Yu, Xifan

A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph

LIPIcs-CCC-2023-30.pdf (1 MB)


We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number p of vertices has value at least Ω(p^{1/3}). This is in contrast to the widely believed conjecture that the actual clique number of the Paley graph is O(polylog(p)). Our result may be viewed as a derandomization of that of Deshpande and Montanari (2015), who showed the same lower bound (up to polylog(p) terms) with high probability for the Erdős-Rényi random graph on p vertices, whose clique number is with high probability O(log(p)). We also show that our lower bound is optimal for the Feige-Krauthgamer construction of pseudomoments, derandomizing an argument of Kelner. Finally, we present numerical experiments indicating that the value of the degree 4 SOS relaxation of the Paley graph may scale as O(p^{1/2 - ε}) for some ε > 0, and give a matrix norm calculation indicating that the pseudocalibration construction for SOS lower bounds for random graphs will not immediately transfer to the Paley graph. Taken together, our results suggest that degree 4 SOS may break the "√p barrier" for upper bounds on the clique number of Paley graphs, but prove that it can at best improve the exponent from 1/2 to 1/3.

BibTeX - Entry

  author =	{Kunisky, Dmitriy and Yu, Xifan},
  title =	{{A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{30:1--30:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-183008},
  doi =		{10.4230/LIPIcs.CCC.2023.30},
  annote =	{Keywords: convex optimization, sum of squares, Paley graph, derandomization}

Keywords: convex optimization, sum of squares, Paley graph, derandomization
Collection: 38th Computational Complexity Conference (CCC 2023)
Issue Date: 2023
Date of publication: 10.07.2023

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