Let H(k,n,p) be the distribution on k-uniform hypergraphs where every subset of [n] of size k is included as an hyperedge with probability p independently. In this work, we design and analyze a simple spectral algorithm that certifies a bound on the size of the largest clique, ω(H), in hypergraphs H ∼ H(k,n,p). For example, for any constant p, with high probability over the choice of the hypergraph, our spectral algorithm certifies a bound of Õ(√n) on the clique number in polynomial time. This matches, up to polylog(n) factors, the best known certificate for the clique number in random graphs, which is the special case of k = 2. Prior to our work, the best known refutation algorithms [Amin Coja-Oghlan et al., 2004; Sarah R. Allen et al., 2015] rely on a reduction to the problem of refuting random k-XOR via Feige’s XOR trick [Uriel Feige, 2002], and yield a polynomially worse bound of Õ(n^{3/4}) on the clique number when p = O(1). Our algorithm bypasses the XOR trick and relies instead on a natural generalization of the Lovász theta semidefinite programming relaxation for cliques in hypergraphs.
@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2022.42, author = {Guruswami, Venkatesan and Kothari, Pravesh K. and Manohar, Peter}, title = {{Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {42:1--42:7}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.42}, URN = {urn:nbn:de:0030-drops-171642}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.42}, annote = {Keywords: Planted clique, Average-case complexity, Spectral refutation, Random matrix theory} }
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