New and Improved Bounds for the Minimum Set Cover Problem

We study the relationship between the approximation factor for the Set-Cover problem and the parameters ∆ : the maximum cardinality of any subset, and k : the maximum number of subsets containing any element of the ground set. We show an LP rounding based approximation of (k− 1)(1− e ln∆ k−1 ) + 1, which is substantially better than the classical algorithms in the range k ≈ ln∆, and also improves on related previous works [19, 22]. For the interesting case when k = θ(log∆) we also exhibit an integrality gap which essentially matches our approximation algorithm. We also prove a hardness of approximation factor of Ω ( log∆ (log log∆)2 ) when k = θ(log∆). This is the first study of the hardness factor specifically for this range of k and ∆, and improves on the only other such result implicitly proved in [18].