Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number

Authors Venkatesan Guruswami, Pravesh K. Kothari, Peter Manohar

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Venkatesan Guruswami
  • University of California Berkeley, CA, USA
Pravesh K. Kothari
  • Carnegie Mellon University, Pittsburgh, PA, USA
Peter Manohar
  • Carnegie Mellon University, Pittsburgh, PA, USA

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Venkatesan Guruswami, Pravesh K. Kothari, and Peter Manohar. Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 42:1-42:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Planted clique
  • Average-case complexity
  • Spectral refutation
  • Random matrix theory


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