Testing the Independence Number of Hypergraphs

Abstract

A k-uniform hypergraph G of size n is said to be ε-far from having an independent set of size ρn if one must remove at least εn ^k edges of G in order for the remaining hypergraph to have an independent set of size ρn. In this work, we present a natural property testing algorithm that distinguishes between hypergraphs which have an independent set of size ≥ ρn and hypergraphs which are ε-far from having an independent set of size ρn. Our algorithm is natural in the sense that we sample $\simeq c(k)\frac{\rho^{2k}}{\varepsilon^3}$ random vertices of G, and according to the independence number of the hypergraph induced by this sample, we distinguish between the two cases above. Here c(k) depends on k alone (e.g. the sample size is independent of n). To the best of our knowledge, property testing of the independence number of hypergraphs has not been addressed in the past.