Counting Matchings of Size k Is $\sharp$ W[1]-Hard

Curticapean, Radu

doi:10.1007/978-3-642-39206-1_30

Cited by 36 publications

(42 citation statements)

References 11 publications

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“…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).…”

Section: Application To Countingmentioning

confidence: 99%

Randomised Enumeration of Small Witnesses Using a Decision Oracle

Meeks

2018

Algorithmica

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Many combinatorial problems involve determining whether a universe of n elements contains a witness consisting of k elements which have some specified property. In this paper we investigate the relationship between the decision and enumeration versions of such problems: efficient methods are known for transforming a decision algorithm into a search procedure that finds a single witness, but even finding a second witness is not so straightforward in general. We show that, if the decision version of the problem can be solved in time f (k) · poly(n), there is a randomised algorithm which enumerates all witnesses in time e k+o(k) · f (k) · poly(n) · N , where N is the total number of witnesses. If the decision version of the problem is solved by a randomised algorithm which may return false negatives, then the same method allows us to output a list of witnesses in which any given witness will be included with high probability. The enumeration algorithm also gives rise to an efficient algorithm to count the total number of witnesses when this number is small.

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Section: Application To Countingmentioning

confidence: 99%

Randomised Enumeration of Small Witnesses Using a Decision Oracle

Meeks

2018

Algorithmica

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“…During the last years, much work has been done in the field of parameterized counting complexity. Important results are the proof of #W [1]-hardness for counting the number of k-matchings in a simple graph [2], and the dichotomies for counting graph homomorphisms [10,4] and embeddings [3].…”

Section: Introductionmentioning

confidence: 99%

Parameterized Counting of Trees, Forests and Matroid Bases

Brand

Roth

2017

Computer Science – Theory and Applications

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We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with k edges are #W[1]-hard when parameterized by k. Together with the recent algorithm for deterministic matrix truncation by Lokshtanov et al. (ICALP 2015), the hardness result for k-forests implies #W[1]-hardness of the problem of counting bases of a matroid when parameterized by rank or nullity, even if the matroid is restricted to be representable over a field of characteristic 2. We complement this result by pointing out that the problem becomes fixed parameter tractable for matroids represented over a fixed finite field.

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“…In this case, it suffices to sample independent sets of size at most k = O(1) according to (1.3), as larger independent sets will have negligible stationary probability. This can be done in O(n k ) time by enumerating all independent sets of size at most k. This is polynomial for constant k, though counting is #W [1]-hard viewed as a fixed parameter problem, even for line graphs [12]. We omit the details here.…”

Section: Preliminariesmentioning

confidence: 99%

Counting Independent Sets in Graphs with Bounded Bipartite Pathwidth

Dyer

Greenhill

Müller

2019

Graph-Theoretic Concepts in Computer Science

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We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity λ, can be viewed as a strong generalisation of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalise line graphs. We consider two further generalisations of claw-free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially-bounded vertex weights. * A preliminary version of this paper appeared as [19]. 1 in graphs for which all bipartite induced subgraphs are well structured, in a sense that we will define precisely. Our approach is to generalise the Markov chain analysis of Jerrum and Sinclair [29] for the corresponding problem of counting matchings.The canonical path argument given by Jerrum and Sinclair in [29] relied on the fact that the symmetric difference of two matchings of a given graph G is a bipartite subgraph of G consisting of a disjoint union of paths and even-length cycles. We introduce a new graph parameter, which we call bipartite pathwidth, to enable us to give the strongest generalisation of the approach of [29], far beyond the class of line graphs.

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Counting Matchings of Size k Is $\sharp$ W[1]-Hard

Cited by 36 publications

References 11 publications

Randomised Enumeration of Small Witnesses Using a Decision Oracle

Randomised Enumeration of Small Witnesses Using a Decision Oracle

Parameterized Counting of Trees, Forests and Matroid Bases

Counting Independent Sets in Graphs with Bounded Bipartite Pathwidth

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