Parameterized Algorithms and Hardness Results for Some Graph Motif Problems
We study the NP-complete Graph Motif problem: given a vertex-colored graph G = (V,E) and a multiset M of colors, does there exist an S ⊆ V such that G[S] is connected and carries exactly (also with respect to multiplicity) the colors in M? We present an improved randomized algorithm for Graph Motif with running time O(4.32|M|·|M|2·|E|). We extend our algorithm to list-colored graph vertices and the case where the motif G[S] needs not be connected. By way of contrast, we show that extending the request for motif connectedness to the somewhat “more robust” motif demands of biconnectedness or bridge-connectedness leads to W-complete problems. Actually, we show that the presumably simpler problems of finding (uncolored) biconnected or bridge-connected subgraphs are W-complete with respect to the subgraph size. Answering an open question from the literature, we further show that the parameter “number of connected motif components” leads to W-hardness even when restricted to graphs that are paths.
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