  The problems studied in this article originate from the Graph Motif problem
introduced by Lacroix et al. in the context of biological networks. The problem
is to decide if a vertex-colored graph has a connected subgraph whose colors
equal a given multiset of colors $M$. It is a graph pattern-matching problem
variant, where the structure of the occurrence of the pattern is not of
interest but the only requirement is the connectedness. Using an algebraic
framework recently introduced by Koutis et al., we obtain new FPT algorithms
for Graph Motif and variants, with improved running times. We also obtain
results on the counting versions of this problem, proving that the counting
problem is FPT if M is a set, but becomes W[1]-hard if M is a multiset with two
colors. Finally, we present an experimental evaluation of this approach on real
datasets, showing that its performance compares favorably with existing
software.
