Abstract
We consider the #W[1]-hard problem of counting all matchings with exactly k edges in a given input graph G; we prove that it remains #W[1]-hard on graphs G that are line graphs or bipartite graphs with degree 2 on one side. In our proofs, we use that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k length-2 paths into (arbitrary) host graphs. Here, a homomorphism from H to G is edge-injective if it maps any two distinct edges of H to distinct edges in G. We show that edge-injective homomorphisms from a pattern graph H can be counted in polynomial time if H has bounded vertex-cover number after removing isolated edges. For hereditary classes $\mathcal {H}$ of pattern graphs, we complement this result: If the graphs in $\mathcal {H}$ have unbounded vertex-cover number even after deleting isolated edges, then counting edge-injective homomorphisms with patterns from $\mathcal {H}$ is #W[1]-hard. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.
Highlights
Since Valiant’s seminal #P-hardness result for the permanent [35], the complexity theory of counting problems has advanced to a classical subfield of computational complexity
We focus on a recent relaxation of hard counting problems by studying their parameterized complexity [18]
The input to a given counting problem comes with a parameter k ∈ N, and we ask whether the problem is fixed-parameter tractable (FPT): That is, can it can be solved in time f (k) · poly(|x|) for some computable function f that may grow super-polynomially? For instance, the #P-complete problem of counting vertex-covers of size k in an n-vertex graph can be solved in time 2k · poly(n) and is FPT [18]
Summary
Since Valiant’s seminal #P-hardness result for the permanent [35], the complexity theory of counting problems has advanced to a classical subfield of computational complexity. For other parameterized problems such as counting cliques, cycles, paths, or matchings of size k in n-vertex graphs, the best known algorithms run in time nO(k), and FPT-algorithms are not believed to exist To substantiate this belief, Flum and Grohe [18] introduced the class #W[1] and identified the problem of counting k-cliques to be complete for #W[1] under parameterized reductions. The decision versions of these last three problems are FPT [1] (or even polynomial-time solvable in the case of matchings) As it turns out, the problem of counting k-matchings plays a central role in parameterized counting. The proofs of lemmas, claims and theorems marked with appear in the full version of this paper, which can be found at http://arxiv.org/abs/1702.05447
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