Parameterized Algorithmics for Finding Connected Motifs in Biological Networks

We study the NP-hard List-Colored Graph Motif problem which, given an undirected list-colored graph G=(V,E) and a multiset M of colors, asks for maximum-cardinality sets S\subseteq V and M^{\prime }\subseteq M such that G[S] is connected and contains exactly (with respect to multiplicity) the colors in M^{\prime }. List-Colored Graph Motif has applications in the analysis of biological networks. We study List-Colored Graph Motif with respect to three different parameterizations. For the parameters motif size \vert M\vert and solution size \vert S\vert, we present fixed-parameter algorithms, whereas for the parameter \vert V\vert -\vert M\vert, we show W[1]-hardness for general instances and achieve fixed-parameter tractability for a special case of List-Colored Graph Motif. We implemented the fixed-parameter algorithms for parameters \vert M\vert and \vert S\vert, developed further speed-up heuristics for these algorithms, and applied them in the context of querying protein-interaction networks, demonstrating their usefulness for realistic instances. Furthermore, we show that extending the request for motif connectedness to stronger demands, such as biconnectedness or bridge-connectedness leads to W[1]-hard problems when the parameter is the motif size \vert M\vert