Generalized block designs as approximations for optimal coverings

Abstract

Let V be a set of cardinality v, v ∈ ‖N. We are looking for the minimal number of k-sets (i.e. subsets of V having cardinality k), such that every t-set of V, t≦k, is covered by at least λ of these k-sets. This special covering problem is called the generalized block design problem with parameters v,k,t,λ. It is equivalent to the problem of Turán [16] and also to the generalized covering problem [4]. Therefore, the known bounds for these two equivalent problems are also bounds for the generalized block design problem and vice versa.

Using some type of greedy algorithm, we will compute an approximative solution for an optimal generalized design with arbitrary parameters. The number of blocks in such an approximation will be at most (1+log( ^k_t ))-times the optimal number of blocks. This result depends essentially on a theorem of Lovász [11].