# Strong inapproximability results on balanced rainbow-colorable hypergraphs

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## Abstract

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 polynomial-time algorithmnds 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 *ϵ*, for any *ϵ* > 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.

One of our main technical tools is based on *reverse hypercontractivity*. Roughly, it says the *noise operator* increases *q*-norm of a function when *q* < 1, which is enough for some special cases of our results. To prove the full results, we generalize the reverse hypercontractivity to more general operators, which might be of independent interest. Together with the generalized reverse hypercontractivity and recent developments in inapproximability based on invariance principles for correlated spaces, we give a *recipe* for converting a promising test distribution and a suitable choice of an outer PCP to hardness of nding 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 rst analytic proof of (*K* − 1 − *ϵ*)-hardness of *K*-Hypergraph Vertex Cover with more structure in completeness, and (2) hardness of (2*Q*+1)-SAT when the input clause is promised to have an assignment where every clause has at least *Q* true literals.

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