SOS Lower Bound for Exact Planted Clique

Author Shuo Pang

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Shuo Pang
  • Mathematics Department, University of Chicago, IL, USA


I am very grateful to Aaron Potechin for the introduction of the problem and the encouraging communications, and to Alexander Razborov for the advice and help on improving the quality of the paper. My thanks also go to the anonymous reviewers for their constructive criticism of the presentation.

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Shuo Pang. SOS Lower Bound for Exact Planted Clique. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 26:1-26:63, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • Mathematics of computing → Random graphs
  • Theory of computation → Semidefinite programming
  • Sum-of-Squares
  • planted clique
  • random graphs
  • average-case lower bound


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