Radu Curticapean

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

For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertex-cover number (equivalently, the size of the maximum matching in C is bounded), then #Sub(C) is polynomial-time solvable. We complement this result with a corresponding lower bound: if C is any recursively enumerable class of graphs with unbounded vertexcover number, then #Sub(C) is #W[1]-hard parameterized by the size of H and hence not polynomial-time solvable and not even fixed-parameter tractable, unless FPT is equal to #W[1]. As a first step of the proof, we show that counting kmatchings in bipartite graphs is #W[1]-hard. Recently, Curticapean [ICALP 2013] 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. As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] stating that counting paths of length k is #W[1]-hard, as well as a similar almost-tight ETH-based lower bound on the exponent.

Counting matchings of size k is # W[1]-hard

We prove

Homomorphisms are a good basis for counting small subgraphs

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Weighted counting of k -matchings is #w[1]-hard

W[1]-hardness of the following parameterized counting problem: Given a simple undirected graph G and a parameter k∈ℕ, compute the number of matchings of size k in G. It is known from [1] that, given an edge-weighted graph G, computing a particular weighted sum over the matchings in G is

Block Interpolation: A Framework for Tight Exponential-Time Counting Complexity

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Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k

W[1]-hard. In the present paper, we exhibit a reduction that does not require weights. This solves an open problem from [5] and adds a natural parameterized counting problem to the scarce list of

Counting Triangulations and Other Crossing-Free Structures via Onion Layers

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The complexity of the cover polynomials for planar graphs of bounded degree

W[1]-hard problems. Since the classical version of this problem is well-studied, we believe that our result facilitates future

A Fixed-Parameter Perspective on #BIS

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Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes

W[1]-hardness proofs for other problems.