Maximum Motif Problem in Vertex-Colored Graphs

Searching for motifs in graphs has become a crucial problem in the analysis of biological networks. In this context, different graph motif problems have been considered [13,7,5]. Pursuing a line of research pioneered by Lacroix et al. [13], we introduce in this paper a new graph motif problem: given a vertex colored graph G and a motif $\mathcal{M}$, where a motif is a multiset of colors, find a maximum cardinality submotif $\mathcal{M}' \subseteq \mathcal{M}$ that occurs as a connected motif in G . We prove that the problem is APX-hard even in the case where the target graph is a tree of maximum degree 3, the motif is actually a set and each color occurs at most twice in the tree. Next, we strengthen this result by proving that the problem is not approximable within factor $2^{\rm {log^{\delta} n}}$, for any constant *** < 1, unless NP *** DTIMEclass(2POLY log n). We complement these results by presenting two fixed-parameter algorithms for the problem, where the parameter is the size of the solution. Finally, we give exact fast exponential-time algorithms for the problem.