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.
@InProceedings{pang:LIPIcs.CCC.2021.26, 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 = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.26}, 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} }
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