Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number
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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 [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 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 Lovasz theta semidefinite programming relaxation for cliques in hypergraphs.
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