The Parameterised Complexity of Counting Connected Subgraphs and Graph Motifs

TL;DR

This paper studies the complexity of counting connected subgraphs and motifs in graphs, showing hardness results and approximation algorithms for these problems within a parameterized complexity framework.

Contribution

It introduces a general class of parameterized counting problems on graphs and establishes complexity and approximation results for problems with bounded treewidth properties.

Findings

Counting connected k-vertex subgraphs is #W[1]-hard.

An FPTRAS exists for certain monotone properties with bounded treewidth minimal graphs.

Results apply to a counting version of the Graph Motif problem.

Abstract

We introduce a class 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 focusses 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.