On parameterized complexity of the Multi-MCS problem

We introduce the maximum common subgraph problem for multiple graphs (Multi-MCS) inspired by various biological applications such as multiple alignments of gene sequences, protein structures, metabolic pathways, or protein-protein interaction networks. Multi-MCS is a generalization of the two-graph Maximum Common Subgraph problem (MCS). On the basis of the framework of parameterized complexity theory, we derive the parameterized complexity of Multi-MCS for various parameters for different classes of graphs. For example, for directed graphs with labeled vertices, we prove that the parameterized m-Multi-MCS problem is W[2]-hard, while the parameterized k-Multi-MCS problem is W[t]-hard (@?t>=1), where m and k are the size of the maximum common subgraph and the number of multiple graphs, respectively. We show similar results for other parameterized versions of the Multi-MCS problem for directed graphs with vertex labels and undirected graphs with vertex and edge labels by giving linear FPT reductions of the problems from parameterized versions of the longest common subsequence problem. Likewise, for unlabeled undirected graphs, we show that a parameterized version of the Multi-MCS problem with a fixed number of input graphs is W[1]-complete by showing a linear FPT reduction to and from a parameterized version of the maximum clique problem.

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