Approximating Independent Set and Coloring in Random Uniform Hypergraphs

Abstract

We consider the problems Independent Set and Coloring in uniform hypergraphs with n vertices. If $\mathcal{NP} \not \subseteq \mathcal{ZPP}$, there are no polynomial worst case running time approximation algorithms with approximation guarantee n ^1 − ε for any ε> 0. We show that the problems are easier to approximate in polynomial expected running time for random hypergraphs. For d ≥ 2, we use the H _d (n,p) model of random d-uniform hypergraphs on n vertices, choosing the edges independently with probability p. We give deterministic algorithms with polynomial expected running time for random inputs from H _d (n,p), and approximation guarantee O(n ^1/2·p ^{− (d − 3)/(2d − 2)}/(ln n)^{1/(d − 1)}).