The parameterised complexity of counting connected subgraphs and graph motifs ?
We introduce a family of parameterised counting problems on graphs, p-#Induced Subgraph With Property(?), which generalises a number of problems which have previously been studied. This paper focuses on the case in which ? defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which ? describes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that there exists an FPTRAS for p-#Induced Subgraph With Property(?) whenever ? is monotone and all the minimal graphs satisfying ? have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem.
parameterised complexity fptras counting complexity