Approximating set multi-covers

Lovász and Stein (independently) proved that any hypergraph satisfies τ ≤ (1+ln ∆)τ * , where τ is the transversal number, τ * is its fractional version, and ∆ denotes the maximum degree. We prove τ f ≤ 3.17τ * max{ln ∆, f } for the f-fold transversal number τ f. Similarly to Lovász and Stein, we also show that this bound can be achieved non-probabilistically, using a greedy algorithm. As a combinatorial application, we prove an estimate on how fast τ f /f converges to τ *. As a geometric application, we obtain a bound on the minimal density of an f-fold covering of the d-dimensional space by translates of any convex body.