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.

翻译：本文针对红蓝集合覆盖问题提出了一种新的近似算法，并给出了新的困难性结果。我们的近似算法实现了$\tilde O(m^{1/3})$-近似，改进了Elkin和Peleg提出的$\tilde O(m^{1/2})$-近似（其中$m$为集合的数量）。此外，我们给出了从最小$k$-并集问题到红蓝集合覆盖问题的近乎保持近似的归约，在密集-随机猜想下，该归约导致了$\tilde\Omega(m^{1/4 - \varepsilon})$的困难性下界。