Parameterized Counting of Trees, Forests and Matroid Bases

Brand, Cornelius; Roth, Marc

doi:10.1007/978-3-319-58747-9_10

Cited by 9 publications

(12 citation statements)

References 22 publications

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“…An example for such a property is the case of Φ(H) = 1 if and only if H is a tree. In contrast, the intractability result for the case of Φ = acyclicity (that is, being a forest) turned out to be much harder to show [12], indicated by connections to parameterized counting problems in matroid theory.…”

Section: Techniquesmentioning

confidence: 99%

“…Given two partitions σ and ρ of a finite set S, we say that σ refines ρ if every block of σ is a subset of a block of ρ, and in this case we write σ ≤ ρ. This induces a partial order, called the partition lattice 12 of S. The explicit formula of the Möbius function over the partition lattice is of particular importance in this work:…”

Section: Möbius Inversion and The Partition Latticementioning

confidence: 99%

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Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders

Roth,

Schmitt,

Wellnitz

2020

Preprint

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Given a graph property Φ, we consider the problem EdgeSub(Φ), where the input is a pair of a graph G and a positive integer k, and the task is to decide whether G contains a k-edge subgraph that satisfies Φ. Specifically, we study the parameterized complexity of EdgeSub(Φ) and of its counting problem #EdgeSub(Φ) with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties Φ: the decision problem EdgeSub(Φ) always admits an FPT ("fixed-parameter tractable") algorithm and the counting problem #EdgeSub(Φ) always admits an FPTRAS ("fixed-parameter tractable randomized approximation scheme"). For exact counting, we present an exhaustive and explicit criterion on the property Φ which, if satisfied, yields fixed-parameter tractability and otherwise #W[1]-hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for #EdgeSub(Φ) that run in time f (k) • |G| o(k/ log k) for any computable function f .As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of #EdgeSub(Φ). This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). This approach does not only apply to #EdgeSub(Φ) but also to the more general problem of computing weighted linear combinations of subgraph counts. As a special case of such a linear combination, we introduce a parameterized variant of the Tutte Polynomial T k G of a graph G, to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, T k G (2, 1) corresponds to the number of k-forests in the graph G. Our techniques allow us to completely understand the parametrized complexity of computing the evaluation of T k G at every pair of rational coordinates (x, y). In particular, our results give a new proof for the #W[1]-hardness of the problem of counting k-forests in a graph.

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Section: Techniquesmentioning

confidence: 99%

Section: Möbius Inversion and The Partition Latticementioning

confidence: 99%

Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders

Roth,

Schmitt,

Wellnitz

2020

Preprint

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“…Given a graph property Φ and a graph H, we write L(Φ, H) for the set of all fractures ρ of H such that H ρ satisfies Φ. Furthermore, the indicator a(Φ, H) of Φ and H is defined as follows: 9 We can even obtain an exponentially small (in |G|) error probability by repeating for O(2 |V (H)| ) • |G| O (1) many trials. Now let G = {G 1 , G 2 , .…”

Section: Fractured Graphsmentioning

confidence: 99%

“…It was shown in [9] that counting k-forests in a graph G is #W[1]-hard. Implicitly, the proof of the latter result also yields a conditional lower bound of f (k) • |G| o(k/ log k) under the Exponential Time Hypothesis.…”

Section: Forests and Linear Forestsmentioning

confidence: 99%

Parameterized (Modular) Counting and Cayley Graph Expanders

Peyerimhoff,

Roth,

Schmitt

et al. 2021

Preprint

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We study the problem #EdgeSub(Φ) of counting k-edge subgraphs satisfying a given graph property Φ in a large host graph G. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients.Our approach relies on novel constructions of low-degree Cayley graph expanders of p-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coefficients in the aforementioned linear combinations over the field Fp which gives us significantly more control over the cancellation behaviour of the coefficients. Our main result is an exhaustive and fine-grained complexity classification of #EdgeSub(Φ) for minor-closed properties Φ, closing the missing gap in previous work by Roth, Schmitt and Wellnitz (ICALP 21).Additionally, we observe that our methods also apply to modular counting. Among others, we obtain novel intractability results for the problems of counting k-forests and matroid bases modulo a prime p. Furthermore, from an algorithmic point of view, we construct algorithms for the problems of counting k-paths and k-cycles modulo 2 that outperform the best known algorithms for their non-modular counterparts.In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime p.

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“…A, relatively small, part of the research on parameterized computation has been focused to the study of parameterized counting problems [40,56,4,16,19,18,17,15,14,21,20,6,9,57,52,51]. Moreover, even less effort has been done for the definition and study of data-reduction concepts for parameterized counting problems.…”

mentioning

confidence: 99%

Compactors for parameterized counting problems

Thilikos

2021

Computer Science Review

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Parameterized Counting of Trees, Forests and Matroid Bases

Cited by 9 publications

References 22 publications

Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders

Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders

Parameterized (Modular) Counting and Cayley Graph Expanders

Compactors for parameterized counting problems

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