Counting Small Induced Subgraphs Satisfying Monotone Properties

Roth, Marc; Schmitt, Johannes; Wellnitz, Philip

doi:10.1109/focs46700.2020.00128

Cited by 7 publications

(40 citation statements)

References 48 publications

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“…We note that our classification for hereditary properties subsumes and strengthens the classification of monotone properties due to Roth, Schmitt and Wellnitz [30] 6 , see Section 1.3.…”

Section: Our Resultssupporting

confidence: 71%

“…Examples of properties invariant under homomorphic equivalence include 5 That is, a conditional lower bound that applies to all Φ is of the form f (k) • |G| o(g(k)) where g is asymptotically smaller than every proper rational function, e.g., g(k) = log(k). 6 However, the classification of properties depending only on the number of edges in [30] is not subsumed.…”

Section: Our Resultsmentioning

confidence: 99%

“…2 Unfortunately, their classification is implicit in the sense that it is not clear how to pinpoint the complexity of #IndSub(Φ) for any concretely given graph property Φ. For this reason, subsequent work focused on establishing explicit criteria for tractability and hardness [11,29,30].…”

Section: Extended Abstractmentioning

confidence: 99%

“…So far, there are only partial results on resolving Conjecture 1 for the hereditary and edge-monotone case; see [30,Section 6] for hereditary properties defined by a single forbidden induced subgraph, and [25,29] for some results on edge-monotone properties. In this work, we study the first case, i.e., hereditary properties; our results are presented below.…”

Section: Extended Abstractmentioning

confidence: 99%

“…Similarly as in previous work [7,11,29,30] we rely on the framework of graph motif parameters and the homomorphism basis as introduced by Curticapean, Dell and Marx [7]: Writing #IndSub(Φ, k → G) for the number of k-vertex induced subgraphs of G that satisfy Φ, it is known that there is a unique function a Φ,k with finite support and independent from G, such that…”

Section: Technical Overviewmentioning

confidence: 99%

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Counting Small Induced Subgraphs with Hereditary Properties

Focke¹,

Roth²

2021

Preprint

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We study the computational complexity of the problem #IndSub(Φ) of counting k-vertex induced subgraphs of a graph G that satisfy a graph property Φ. Our main result establishes an exhaustive and explicit classification for all hereditary properties, including tight conditional lower bounds under the Exponential Time Hypothesis (ETH):• If a hereditary property Φ is true for all graphs, or if it is true only for finitely many graphs, then #IndSub(Φ) is solvable in polynomial time.• Otherwise, #IndSub(Φ) is #W[1]-complete when parameterised by k, and, assuming ETH, it cannot be solved in timeThis classification features a wide range of properties for which the corresponding detection problem (as classified by Khot and Raman [TCS 02]) is tractable but counting is hard. Moreover, even for properties which are already intractable in their decision version, our results yield significantly stronger lower bounds for the counting problem.As additional result, we also present an exhaustive and explicit parameterised complexity classification for all properties that are invariant under homomorphic equivalence.By covering one of the most natural and general notions of closure, namely, closure under vertexdeletion (hereditary), we generalise some of the earlier results on this problem. For instance, our results fully subsume and strengthen the existing classification of #IndSub(Φ) for monotone (subgraph-closed) properties due to Roth, Schmitt, and Wellnitz [FOCS 20].

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“…We note that our classification for hereditary properties subsumes and strengthens the classification of monotone properties due to Roth, Schmitt and Wellnitz [30] 6 , see Section 1.3.…”

Section: Our Resultssupporting

confidence: 71%

Section: Our Resultsmentioning

confidence: 99%

Section: Extended Abstractmentioning

confidence: 99%

Section: Extended Abstractmentioning

confidence: 99%

Section: Technical Overviewmentioning

confidence: 99%

See 3 more Smart Citations

Counting Small Induced Subgraphs with Hereditary Properties

Focke¹,

Roth²

2021

Preprint

Self Cite

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Exact and Approximate Pattern Counting in Degenerate Graphs: New Algorithms, Hardness Results, and Complexity Dichotomies

Bressan

Roth

2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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Parameterized Counting and Cayley Graph Expanders

Peyerimhoff

Roth

Schmitt

et al. 2023

SIAM J. Discrete Math.

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Given a graph property \Phi , we consider the problem \# EdgeSub(\Phi ), where the input is a pair of a graph G and a positive integer k, and the task is to compute the number of k-edge subgraphs in G that satisfy \Phi . Specifically, we study the parameterized complexity of \#EdgeSub(\Phi ) with respect to both approximate and exact counting, as well as its decision version EdgeSub(\Phi ). Among others, our main result fully resolves the case of minor-closed properties \Phi : the decision problem EdgeSub(\Phi ) always admits a fixed-parameter tractable algorithm, and the counting problem \#EdgeSub(\Phi ) always admits a fixed-parameter tractable randomized approximation scheme. For exact counting, we present an exhaustive and explicit criterion on the property \Phi which, if satisfied, yields fixed-parameter tractability and otherwise \#\sansW [\sansone ]-hardness. Additionally, our hardness results come with an almost tight conditional lower bound under the exponential time hypothesis. Our main technical result concerns the exact counting problem: Building upon the breakthrough result of Curticapean, Dell, and Marx (Symposium on Theory of Computing 2017), we express the number of subgraphs satisfying \Phi 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 analyze the coefficients in the aforementioned linear combinations over the field \BbbF p which gives us significantly more control over the cancelation behavior of the coefficients.

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Counting Small Induced Subgraphs Satisfying Monotone Properties

Cited by 7 publications

References 48 publications

Counting Small Induced Subgraphs with Hereditary Properties

Counting Small Induced Subgraphs with Hereditary Properties

Exact and Approximate Pattern Counting in Degenerate Graphs: New Algorithms, Hardness Results, and Complexity Dichotomies

Parameterized Counting and Cayley Graph Expanders

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