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, for example, APX-hardness even if the input graph is a properly edge-colored path in which every color occurs at most twice. Moreover, we show that RAINBOW SUBGRAPH is W[1]-hard with respect to the parameter k and also with respect to the dual parameter l := 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 l and "maximum number of colors incident with any vertex".