Strong Inapproximability Results on Balanced Rainbow-Colorable Hypergraphs

Consider a K-uniform hypergraph H = (V,E). A coloring c : V → {1, 2, . . . , k} with k colors is rainbow if every hyperedge e contains at least one vertex from each color, and is called perfectly balanced when each color appears the same number of times. A simple polynomialtime algorithm finds a 2-coloring if H admits a perfectly balanced rainbow k-coloring. For a hypergraph that admits an almost balanced rainbow coloring, we prove that it is NP-hard to find an independent set of size , for any > 0. Consequently, we cannot weakly color (avoiding monochromatic hyperedges) it with O(1) colors. With k = 2, it implies strong hardness for discrepancy minimization of systems of bounded set-size. Our techniques extend recent developments in inapproximability based on reverse hypercontractivity and invariance principles for correlated spaces. We give a recipe for converting a promising test distribution and a suitable choice of a outer PCP to hardness of finding an independent set in the presence of highly-structured colorings. We use this recipe to prove additional results almost in a black-box manner, including: (1) the first analytic proof of (K−1− )hardness of K-Hypergraph Vertex Cover with more structure in completeness, and (2) hardness of (2Q + 1)-SAT when the input clause is promised to have an assignment where every clause has at least Q true literals.