Uniform Random Covering Problems

Covering Non-uniform Hypergraphs

A subset of the vertices in a hypergraph is a cover if it intersects every edge. Let τ(H) denote the cardinality of a minimum cover in the hypergraph H , and let us denote by g(n) the maximum of τ(H) taken over all hypergraphs H with n vertices and with no two hyperedges of the same size. We show that g(n) < 1.98 √ n(1 + o(1)). A special case corresponds to an old problem of Erdős asking the ma...

Covering Problems

Random Covering Designs

A t (n; k; ) covering design (n k > t 2) consists of a collection of k-element subsets (blocks) of an n-element set X such that each t-element subset of X occurs in at least blocks. Let = 1 and k 2t 1. Consider a randomly selected collection B of blocks; jBj = (n). We use the correlation inequalities of Janson ([10], [1]) to show that B exhibits a rather sharp threshold behaviour, in the sense ...

Min-Power Covering Problems

In the classical vertex cover problem, we are given a graph G = (V,E) and we aim to find a minimum cardinality cover of the edges, i.e. a subset of the vertices C ⊆ V such that for every edge e ∈ E, at least one of its extremities belongs to C. In the Min-Power-Cover version of the vertex cover problem, we consider an edge-weighted graph and we aim to find a cover of the edges and a valuation (...

Probabilistic covering problems