Hardness results for coloring 3-colorable 3-uniform hypergraphs

In this paper, we consider the problem of coloring a 3-colorable 3-uniform hypergraph. In the minimization version of this problem, given a 3-colorable 3-uniform hypergraph, one seeks an algorithm to color the hypergraph with as few colors as possible. We show that it is NP-hard to color a 3-colorable 3-uniform hypergraph with constantly many colors. In fact, we show a stronger result that it is NP-hard to distinguish whether a 3-uniform hypergraph with n vertices is 3-colorable or it contains no independent set of size deltan for an arbitrarily small constant delta > 0. In the maximization version of the problem, given a 3-uniform hypergraph, the goal is to color the vertices with 3 colors so as to maximize the number of non-monochromatic edges. We show that it is NP-hard to distinguish whether a 3-uniform hypergraph is 3-colorable or any coloring of the vertices with 3 colors has at most (8)/(9) + epsilon fraction of the edges non-monochromatic where delta > 0 is an arbitrarily small constant. This result is tight since assigning a random color independently to every vertex makes (8)/(9) fraction of the edges non-monochromatic. These results are obtained via a new construction of a probabilistically checkable proof system (PCP) for NP We develop a new construction of the PCP Outer Verifier. An important feature of this construction is smoothening of the projection maps. Dinur, Regev and Smyth [6] independently showed that it is NP-hard to color a 2-colorable 3-uniform hypergraph with constantly many colors. In the "good case", the hypergraph they construct is 2-colorable and hence their result is stronger In the "bad case" however, the hypergraph we construct has a stronger property, namely, it does not even contain an independent set of size deltan.