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.APPROX/RANDOM.2022.42
URN: urn:nbn:de:0030-drops-171642
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17164/
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Guruswami, Venkatesan ; Kothari, Pravesh K. ; Manohar, Peter

Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number

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Abstract

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.

BibTeX - Entry

@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/opus/volltexte/2022/17164},
  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}
}

Keywords: Planted clique, Average-case complexity, Spectral refutation, Random matrix theory
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)
Issue Date: 2022
Date of publication: 15.09.2022

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