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