On Constrained Hypergraph Coloring and Scheduling

We consider the design of approximation algorithms for multicolor generalization of the well known hypergraph 2-coloring problem (property B). Consider a hypergraph H with n vertices, s edges, maximum edge degree D(≤ s) and maximum vertex degree d(≤ s). We study the problem of coloring the vertices of H with minimum number of colors such that no hyperedge i contains more than bi vertices of any color. The main result of this paper is a deterministic polynomial time algorithm for constructing approximate, d(1 + )OPT e-colorings ( ∈ (0, 1)) satisfying all constraints provided that bi’s are logarithmically large in d and two other parameters. This approximation ratio is independent of s. Our lower bound on the bi’s is better than the previous best bound. Due to the similarity of structure these methods can also be applied to resource constrained scheduling. We observe, using the non-approximability result for graph coloring of Feige and Killian[4], that unless NP ⊆ ZPP we cannot find a solution with approximation ratio s 1 2 −δ in polynomial time, for any fixed small δ > 0.