Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number

TL;DR

This paper introduces a spectral algorithm that certifies tight bounds on the clique number in random hypergraphs, surpassing previous methods by avoiding the XOR trick and using a generalized Lovasz theta relaxation.

Contribution

The work presents a novel spectral algorithm for hypergraph clique certification that improves bounds and bypasses traditional XOR-based refutation techniques.

Findings

Certifies a  O(\u221A n) bound on clique size in hypergraphs.

Matches the best known bounds for random graphs up to polylog factors.

Outperforms previous algorithms relying on XOR refutations.

Abstract

Let $\mathcal{H}(k,n,p)$ 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, $\omega(H)$, in hypergraphs $H \sim \mathcal{H}(k,n,p)$. For example, for any constant $p$, with high probability over the choice of the hypergraph, our spectral algorithm certifies a bound of $\tilde{O}(\sqrt{n})$ on the clique number in polynomial time. This matches, up to $\textrm{polylog}(n)$ 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 [CGL04, AOW15] rely on a reduction to the problem of refuting random $k$-XOR via Feige's XOR trick [Fei02], and yield a polynomially worse bound of $\tilde{O}(n^{3/4})$ on the clique number when $p = O(1)$. Our algorithm bypasses the XOR trick and relies instead on a natural generalization of the Lovasz theta semidefinite programming relaxation for cliques in hypergraphs.