Approximating Red-Blue Set Cover

We provide a new approximation algorithm for the Red-Blue Set Cover problem and give a new hardness result. Our approximation algorithm achieves $\tilde O(m^{1/3})$-approximation improving on the $\tilde O(m^{1/2})$-approximation due to Elkin and Peleg (where $m$ is the number of sets). Additionally, we provide a nearly approximation preserving reduction from Min $k$-Union to Red-Blue Set Cover that gives an $\tilde\Omega(m^{1/4 - \varepsilon})$ hardness under the Dense-vs-Random conjecture.