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(Φ), which generalises a number of problems which have previously been studied. This paper focuses on the case in which Φ defines a family of graphs whose edge-minimal elements all have bounded treew...

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Published in:	Journal of computer and system sciences Vol. 81; no. 4; pp. 702 - 716
Main Authors:	Jerrum, Mark, Meeks, Kitty
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01.06.2015
Subjects:	Complexity Counting Counting complexity FPTRAS Graphs Parameterised complexity Parameterised complexity Counting complexity FPTRAS
ISSN:	0022-0000, 1090-2724
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Abstract	We introduce a family of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focuses on the case in which Φ defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φ 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(Φ) whenever Φ is monotone and all the minimal graphs satisfying Φ have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem.
AbstractList	We introduce a family of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focuses on the case in which Φ defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φ 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(Φ) whenever Φ is monotone and all the minimal graphs satisfying Φ have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem. We introduce a family of parameterised counting problems on graphs, p-#Induced Subgraph With Property( Phi ), which generalises a number of problems which have previously been studied. This paper focuses on the case in which Phi defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Phi 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( Phi ) whenever Phi is monotone and all the minimal graphs satisfying Phi have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem.
Author	Meeks, Kitty Jerrum, Mark
Author_xml	– sequence: 1 givenname: Mark surname: Jerrum fullname: Jerrum, Mark email: m.jerrum@qmul.ac.uk – sequence: 2 givenname: Kitty surname: Meeks fullname: Meeks, Kitty email: kitty.meeks@glasgow.ac.uk
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SubjectTerms	Complexity Counting Counting complexity FPTRAS Graphs Parameterised complexity
Title	The parameterised complexity of counting connected subgraphs and graph motifs
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