The parameterised complexity of counting connected subgraphs and graph motifs

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 focusses 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.

Randomised Enumeration of Small Witnesses Using a Decision Oracle

“…However, in the case that the number N of witnesses is large, an enumeration algorithm necessarily takes time at least (N ), whereas we might hope for much better if our goal is simply to determine the total number of witnesses. The family of self-contained k-witness problems studied here includes subgraph problems, whose parameterised complexity from the point of view of counting has been a rich topic for research in recent years [10,11,14,[17][18][19]22]. Many such counting problems, including those whose decision problem belongs to FPT, are known to be #W [1]-complete (see [15] for background on the theory of parameterised counting complexity).…”

“…Positive results in this setting typically exploit structural properties of the graphs involved (e.g. small treewidth) to design (approximate) counting algorithms for inputs with these properties, avoiding any dependence on N [2,3,18].…”

Some Hard Families of Parameterized Counting Problems

“…The classes of counting problems we consider fall within the scope of the general model introduced in [11]; this model describes parameterised counting problems in which the goal is to count labelled subgraphs with particular properties. We repeat the definition here for completeness, before extending it to colourful subgraph counting problems (which we will need for intermediate stages in our reductions).…”

“…A number of these results concern the complexity of induced subgraph counting problems: Chen and Flum [2] demonstrated that problems of counting k-vertex induced paths and of counting k-vertex induced cycles are both #W [1]-complete, and more generally Chen, Thurley and Weyer [3] showed that it is #W [1]-complete to count the number of induced subgraphs isomorphic to a given graph from the class C (p-#Induced Subgraph Isomorphism(C)) whenever C contains arbitrarily large graphs. Other results concern the complexity of "non-induced" subgraph counting problems, including the problems of counting the number of paths (p-#Path) and cycles (p-#Cycle) [9], matchings (p-#Matching [4]), and connected subgraphs (p-#Connected Induced Subgraph [11]); the well-studied problem of counting the number of k-vertex cliques (p-#Clique [9]) can be considered as either an induced or non-induced subgraph problem. However, even considering these examples, the number of problems known to be complete for the parameterised complexity class #W [1] as a whole remains relatively small.…”

“…The precise formulation of our results makes use of the general model for parameterised subgraph counting problems introduced in [11] and described in Section 1.4 below, but informally we show that counting labelled induced subgraphs with property Φ is #W [1]-complete in each of the following situations:…”