Counting small induced subgraphs with hereditary properties

Focke, Jacob; Roth, Marc

doi:10.1145/3519935.3520008

“…This problem seems to be much harder. It is conjectured by Floderus, Kowaluk, Lingas, Lundell [9] that counting induced subgraphs for any k-vertex pattern graph is as hard as counting k-cliques for sufficiently large k. Several works have considered the parameterized complexity of counting subgraphs (See [5,4,17,10,7]) where the primary goal is to obtain a dichotomy of easy vs hard based on structural graph parameters. Some works have also considered restrictions on host graphs such as d-degeneracy [3].…”

Section: Related Workmentioning

confidence: 99%

Finding and Counting Patterns in Sparse Graphs

MISHRA,

Komarath,

Kumar

et al. 2024

Preprint

0

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“…Curticapean [22] further showed that counting matchings with k edges in a graph is also complete for #W [1]. These results led to several remarkable completeness results and new techniques (see, e.g., the works of Curticapean [23,24], Curticapean, Dell and Marx [25], Jerrum and Meeks [26], Brand and Roth [27], as well as recent advances [28,29]).…”

Section: Introductionmentioning

confidence: 99%

Parameterised Counting in Logspace

Haak

¹

,

Meier

²

,

Prakash

³

et al. 2023

Algorithmica

0

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Logarithmic space-bounded complexity classes such as $$\textbf{L} $$ L and $$\textbf{NL} $$ NL play a central role in space-bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space-bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space-bounded models developed by Elberfeld, Stockhusen and Tantau. They defined the operators $$\textbf{para}_{\textbf{W}}$$ para W and $$\textbf{para}_\beta $$ para β for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators $$\textbf{para}_{\textbf{W}}$$ para W and $$\textbf{para}_\beta $$ para β by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, $$\textbf{para}_{{\textbf{W}}[1]}$$ para W [ 1 ] and $$\textbf{para}_{\beta {\textbf{tail}}}$$ para β tail . Then, we consider counting versions of all four operators and apply them to the class $$\textbf{L} $$ L . We obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0, 1)-matrices is $$\#\textbf{para}_{\beta {\textbf{tail}}}\textbf{L} $$ # para β tail L -hard and can be written as the difference of two functions in $$\#\textbf{para}_{\beta {\textbf{tail}}}\textbf{L} $$ # para β tail L . These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of $$\#\textbf{para}_{\beta {\textbf{tail}}}\textbf{L} $$ # para β tail L under parameterised logspace parsimonious reductions coincides with $$\#\textbf{para}_\beta \textbf{L} $$ # para β L . In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research.

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“…There is also a multi-coloured variant of the problem, MISWP(Φ), in which every vertex in G receives one of k colours, and we seek a k-vertex induced subgraph with property Φ and one vertex of each colour. See [43] for a survey of results on exact and approximate #ISWP(Φ) and #MISWP(Φ), and [27] for a more recent complexity classification of #ISWP(Φ) whenever Φ is a hereditary property. Observe that ISWP is a uniform witness problem, where the witnesses are the k-vertex subsets of V (G) which induce copies of graphs satisfying Φ, and Colourful-ISWP(Φ) is simply MISWP(Φ); hence Theorem 1.7 immediately implies that there is an FPTRAS for #MISWP(Φ) and #ISWP(Φ) whenever there is an FPT decision algorithm for MISWP(Φ), and moreover that the running times of these algorithms are the same up to a sub-polynomial factor in the instance size.…”

Section: Introductionmentioning

confidence: 99%

Approximately Counting and Sampling Small Witnesses Using a Colorful Decision Oracle

Dell¹,

Lapinskas²,

Meeks³

2022

SIAM J. Comput.

4

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In this paper, we design efficient algorithms to approximately count the number of edges of a given k-hypergraph, and to sample an approximately uniform random edge. The hypergraph is not given explicitly, and can be accessed only through its colourful independence oracle:The colourful independence oracle returns yes or no depending on whether a given subset of the vertices contains an edge that is colourful with respect to a given vertex-colouring. Our results extend and/or strengthen recent results in the graph oracle literature due to Beame et al. (ITCS 2018), Dell and Lapinskas (STOC 2018), and Bhattacharya et al. (ISAAC 2019).Our results have consequences for approximate counting/sampling: We can turn certain kinds of decision algorithms into approximate counting/sampling algorithms without causing much overhead in the running time. We apply this approximate-counting/sampling-to-decision reduction to key problems in fine-grained complexity (such as k-SUM, k-OV and weighted k-Clique) and parameterised complexity (such as induced subgraphs of size k or weight-k solutions to CSPs).

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Counting small induced subgraphs with hereditary properties

Cited by 5 publications

References 29 publications

Finding and Counting Patterns in Sparse Graphs

Finding and Counting Patterns in Sparse Graphs

Parameterised Counting in Logspace

Approximately Counting and Sampling Small Witnesses Using a Colorful Decision Oracle

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