eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-09-15
42:1
42:7
10.4230/LIPIcs.APPROX/RANDOM.2022.42
article
Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number
Guruswami, Venkatesan
1
Kothari, Pravesh K.
2
Manohar, Peter
2
University of California Berkeley, CA, USA
Carnegie Mellon University, Pittsburgh, PA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol245-approx-random2022/LIPIcs.APPROX-RANDOM.2022.42/LIPIcs.APPROX-RANDOM.2022.42.pdf
Planted clique
Average-case complexity
Spectral refutation
Random matrix theory