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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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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ISSN:0022-0000
1090-2724
DOI:10.1016/j.jcss.2014.11.015