The problems studied in this article originate in the Graph Motif problem
introduced by Lacroix et al. (TCBB 2006) in the context of biological networks.
The problem is to decide if a vertex-colored graph has a connected subgraph
whose colors equals a given multiset of colors M. Using an algebraic framework
recently introduced in Koutis (ICALP 2008) and Koutis, Williams (ICALP 2009),
we obtain new FPT algorithms for Graph Motif and variants, with improved
running times.