Complexity of Counting Subgraphs: Only the Boundedness of the Vertex-Cover Number Counts

Abstract

For a class H of graphs, #Sub(H) is the counting problem that, given a graph H ∈ H and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if H has bounded vertex-cover number (equivalently, the size of the maximum matching in H is bounded), then #Sub(H) is polynomial-time solvable. We complement this result with a corresponding lower bound: if H is any recursively enumerable class of graphs with unbounded vertex-cover number, then #Sub(H) is #W[1]-hard parameterized by the size of H and hence not polynomial-time solvable and not even fixed-parameter tractable, unless FPT = #W [1].As a first step of the proof, we show that counting k-matchings in bipartite graphs is #W[1]-hard. Recently, Curticapean [ICALP 2013] [16] proved the #W[1]-hardness of counting k-matchings in general graphs; our result strengthens this statement to bipartite graphs with a considerably simpler proof and even shows that, assuming the Exponential Time Hypothesis (ETH), there is no f (k)n o(k/ log k) time algorithm for counting k-matchings in bipartite graphs for any computable function f (k). As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] [23] stating that counting paths of length k is #W[1]-hard, as well as a similar almosttight ETH-based lower bound on the exponent.

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).…”

Homomorphisms are a good basis for counting small subgraphs

“…Given these modest improvements for counting subgraph patterns of unbounded vertex-cover number, it is tempting to conjecture that "the exponent cannot remain constant" for such patterns. A result by a subset of the authors [17] shows that this conjecture is indeed true-for an appropriate formalization of the respective computational problem and under appropriate complexity-theoretic assumptions.…”

“…The parameterized reduction in [17] was very complex with various special cases and a Ramsey argument that made g a very large function. While it was sufficient to conditionally rule out f (k) · n c time algorithms for #Sub(H) for any constant c and graph class H of unbounded vertex-cover number, it left open the possibility of, for example, n √ vc(H) time algorithms.…”

“…[40,17]). Essentially, one can exhaustively iterate over the image of the minimum vertex-cover in G, which gives rise to n vc(G) choices; the rest of H can then be embedded by dynamic programming.…”

Some Hard Families of Parameterized Counting Problems

“…1 non-trivial in this sense means that there is no constant c so that, for any k ∈ N, we can determine whether a given graph has the desired property by examining only edges incident with some fixed set of c vertices (a dichotomy result for a special class of these problems was very recently proved by Curticapean and Marx [5], in which parameterised tractability coincides exactly with this definition of triviality). 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.…”