Approximating set multi-covers

Abstract: Johnson and Lov\'asz and Stein proved independently that any hypergraph satisfies $\tau\leq (1+\ln \Delta)\tau^{\ast}$, where $\tau$ is the transversal number, $\tau^{\ast}$ is its fractional version, and $\Delta$ denotes the maximum degree. We prove $\tau_f\leq c \tau^{\ast}\max{\ln \Delta, f}$ for the $f$-fold transversal number $\tau_f$. Similarly to Johnson, Lov\'asz 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 $\tau_f/f$ converges to $\tau^{\ast}$. As a geometric application, we obtain an upper bound on the minimal density of an $f$-fold covering of the $d$-dimensional Euclidean space by translates of any convex body.