The Parameterized Complexity of the Rainbow Subgraph Problem

The NP-hard RAINBOW SUBGRAPH problem, motivated from bioinformatics, is to find in an edge-colored graph a subgraph that contains each edge color exactly once and has at most $k$ vertices. We examine the parameterized complexity of RAINBOW SUBGRAPH for paths, trees, and general graphs. We show that RAINBOW SUBGRAPH  is W[1]-hard with respect to the parameter $k$ and also with respect to the dual parameter $\ell:=n-k$ where $n$ is the number of vertices. Hence, we examine parameter combinations and show, for example, a polynomial-size problem kernel for the combined parameter $\ell$ and ``maximum number of colors incident with any vertex''. Additionally, we show APX-hardness even if the input graph is a properly edge-colored path in which every color occurs at most twice