Hardness of Approximate Hypergraph Coloring

We introduce the notion of covering complexity of a probabilistic verifier. The covering complexity of a verifier on a given input is the minimum number of proofs needed to “satisfy” the verifier on every random string, i.e., on every random string, at least one of the given proofs must be accepted by the verifier. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems, and in particular, (hyper)-graph coloring problems. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a super-constant covering complexity for statements not in the language. Moreover, the acceptance predicate of this verifier is a simple Not-all-Equal check on the four bits it reads. This enables us to prove that for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors, and also yields a super-constant inapproximability result under a stronger hardness assumption.