Streaming Algorithms for 2-Coloring Uniform Hypergraphs

Abstract

We consider the problem of two-coloring n-uniform hypergraphs. It is known that any such hypergraph with at most $\frac{1}{10}\sqrt{\frac{n}{\ln n}} 2^n$ hyperedges can be two-colored [7]. In fact, there is an efficient (requiring polynomial time in the size of the input) randomized algorithm that produces such a coloring. As stated [7], this algorithm requires random access to the hyperedge set of the input hypergraph. In this paper, we show that a variant of this algorithm can be implemented in the streaming model (with just one pass over the input), using space O(|V|B), where V is the vertex set of the hypergraph and each vertex is represented by B bits. (Note that the number of hyperedges in the hypergraph can be superpolynomial in |V|, and it is not feasible to store the entire hypergraph in memory.)

We also consider the question of the minimum number of hyperedges in non-two-colorable n-uniform hypergraphs. Erdős showed that there exist non-2-colorable n-uniform hypegraphs with O(n ² 2ⁿ) hyperedges and Θ(n ²) vertices. We show that the choice Θ(n ²) for the number of vertices in Erdös’s construction is crucial: any hypergraph with at most $\frac{2 n^2}{t}$ vertices and $2^n\exp(\frac{t}{8})$ hyperedges is 2-colorable. (We present a simple randomized streaming algorithm to construct the two-coloring.) Thus, for example, if the number of vertices is at most n ^1.5, then any non-2-colorable hypergraph must have at least $2^n \exp(\frac{\sqrt{n}}{8}) \gg n^22^n$ hyperedges. We observe that the exponential dependence on t in our result is optimal up to constant factors.