Abstract

In this paper, we consider the problem of coloring a 3-colorable 3-uniform hypergraph. In the min-imization version of this problem, given a 3-colorable 3-uniform hypergraph, one seeks an algorithm tocolor 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 independentset of size ffin for an arbitrarily small constant ffi? 0. In the maximization version of the problem, givena 3-uniform hypergraph, the goal is to color the vertices with 3 colors so as to maximize the number ofnon-monochromatic edges. We show that it is NP-hard to distinguish whether a 3-uniform hypergraphis 3-colorable or any coloring of the vertices with 3 colors has at most 89 + ffl fraction of the edges non-monochromatic where ffl? 0 is an arbitrarily small constant. This result is tight since assigning a randomcolor 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 construc-tion is smoothening of the projection maps. We believe that the techniques in this paper would be quite useful in future. As an application of ourtechniques, we give a simpler proof of H*astad's result [11] that for every constant ffl? 0, it is NP-hardto distinguish satisfiable instances of Max-3SAT from instances where no assignment satisfies more that 78 + ffl fraction of the clauses. Dinur, Regev and Smyth [6] independently showed that it is NP-hard to color a 2-colorable 3-uniformhypergraph with constantly many colors. In the "good case", the hypergraph they construct is 2-colorableand 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 ffin.

Citations