Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness

Roth, Marc; Schmitt, Johannes

doi:10.1007/s00453-020-00676-9

Cited by 11 publications

(30 citation statements)

References 28 publications

(96 reference statements)

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“…14 The complexity of computing an ε-approximation of #IndSub(Φ) was investigated by Jerrum and Meeks in a sequence of papers [19][20][21]25], and in case of hereditary properties, it was ultimately resolved by 13 In the definition we can even get away with only considering graphs whose twin-free quotient contains at least two vertices. This is a technicality which makes the class of covered properties more general and ensures that this result covers, for instance, also the property of being (dis)connected, which was of interest in some of the earlier works [19,29]. 14 We remark that our notion of meagre coincides with their notion of meagre in the special case where Φ is true for at most one k-vertex graph for each k, which applies, e.g., to the properties of being a path, a cycle, or a matching.…”

Section: Further Related Workmentioning

confidence: 92%

“…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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Section: Further Related Workmentioning

confidence: 92%

Section: Extended Abstractmentioning

confidence: 99%

Section: Extended Abstractmentioning

confidence: 99%

Section: Technical Overviewmentioning

confidence: 99%

See 2 more Smart Citations

Counting Small Induced Subgraphs with Hereditary Properties

Focke¹,

Roth²

2021

Preprint

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“…Initiated by the breakthrough result by Curticapean, Dell, and Marx [16], a line of research [52,53,20] lifted the dichotomy of Dalmau and Jonnson [19] to all parameterized counting problems that can be expressed as linear combinations of homomorphisms, subsuming counting of subgraphs, counting of induced subgraphs and even counting of answers to existential first-order queries. This lifting technique is sometimes also called complexity monotonicity.…”

Section: The Doubly Restricted Version Of Counting Homomorphismsmentioning

confidence: 99%

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory

Roth

Wellnitz

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Counting homomorphisms from a graph H into another graph G is a fundamental problem of (parameterized) counting complexity theory. In this work, we study the case where both graphs H and G stem from given classes of graphs: H ∈ H and G ∈ G. By this, we combine the structurally restricted version of this problem (where the class G = is the set of all graphs), with the languagerestricted version (where the class H = is the set of all graphs). The structurally restricted version allows an exhaustive complexity classification for classes H: Either we can count all homomorphisms in polynomial time (if the treewidth of H is bounded), or the problem becomes #W[1]-hard [Dalmau, Jonsson, Th.Comp.Sci'04]. In contrast, in this work, we show that the combined view most likely does not admit such a complexity dichotomy.Our main result is a construction based on Kneser graphs that associates every problem P in #W[1] with two classes of graphs H and G such that the problem P is equivalent to the problem #Hom(H → G) of counting homomorphisms from a graph in H to a graph in G. In view of Ladner's seminal work on the existence of NP-intermediate problems [J.ACM'75] and its adaptations to the parameterized setting, a classification of the class #W[1] in fixed-parameter tractable and #W[1]complete cases is unlikely. Hence, obtaining a complete classification for the problem #Hom(H → G) seems unlikely. Further, our proofs easily adapt to W[1] and the problem of deciding whether a homomorphism between graphs exists.In search of complexity dichotomies, we hence turn to special graph classes. Those classes include line graphs, claw-free graphs, perfect graphs, and combinations thereof, and F -colorable graphs for fixed graphs F : If the class G is one of those classes and the class H is closed under taking minors, then we establish explicit criteria for the class H that partition the family of problems #Hom(H → G) into polynomial-time solvable and #W[1]-hard cases. In particular, we can drop the condition of H being minor-closed for F -colorable graphs. As a consequence, we are able to lift the framework of graph motif parameters due to Curticapean, Dell and Marx [STOC'17] to F -colorable graphs and provide an exhaustive classification for the parameterized subgraph counting problem on F -colorable graphs. As a special case, we obtain an easy proof of the parameterized intractability result of the problem of counting k-matchings in bipartite graphs.

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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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Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness

Cited by 11 publications

References 28 publications

Counting Small Induced Subgraphs with Hereditary Properties

Counting Small Induced Subgraphs with Hereditary Properties

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory

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

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