You are given a graph $G = (V,E)$ with $n$ vertices. It might be bipartite if you  want. There are $m$ sets of edges $E_1,\ldots, E_m \subseteq E$ (say disjoint). I am interested in the problem of finding a subset $S \subseteq V$, as small as possible (or even smaller), such that the induced graph $G_S$ has at least one edge from each class $E_i$, for $i=1,\ldots, m$. Currently, I know that this problem is set cover hard. I also have a not completely obvious (roughly) $O(\sqrt{n})$ approximation. This seems like a natural problem - is anyone aware of any relevant references, or any better algorithms?
Finding good induced subgraph
ds.algorithms;graph theory;optimization
Look for Minimum Rainbow Subgraph.