The parameterised complexity of counting even and odd induced subgraphs

Jerrum, Mark; Meeks, Kitty

doi:10.1007/s00493-016-3338-5

Cited by 17 publications

(33 citation statements)

References 10 publications

(21 reference statements)

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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%

“…The issue of finding a single witness using an oracle for the decision problem has previously been investigated by Björklund et al [6], motivated by the fact that the fastest known parameterised algorithms for a number of widely studied problems (such as graph motif [5] and k-path [4]) are non-constructive in nature. Moreover, for some problems (such as k-Clique or Independent Set [3] and k-Even Subgraph [17]) the only known FPT decision algorithm relies on a Ramsey theoretic argument which says the answer must be "yes" provided that the input graph avoids certain easily recognisable structures.…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

Randomised Enumeration of Small Witnesses Using a Decision Oracle

Meeks

2018

Algorithmica

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“…Let us also address the problem of counting small induced subgraphs from a class H. This is a natural and well-studied variant of counting subgraph copies [38,13,33,34,35,45], and for several applications it represents a more appropriate notion of "pattern occurrence". From the perspective of dichotomy results however, it is less intricate than subgraphs or homomorphisms: Counting induced subgraphs is known to be #W [1] [33,34,35,45] introduced the following generalization of the problems #Ind(H) to fixed graph properties Φ: Given a graph G and k ∈ N, the task is to compute the number of induced k-vertex subgraphs that have property Φ.…”

Section: Counting Small Induced Subgraphsmentioning

confidence: 99%

“…From the perspective of dichotomy results however, it is less intricate than subgraphs or homomorphisms: Counting induced subgraphs is known to be #W [1] [33,34,35,45] introduced the following generalization of the problems #Ind(H) to fixed graph properties Φ: Given a graph G and k ∈ N, the task is to compute the number of induced k-vertex subgraphs that have property Φ. Let us call this problem #IndProp(Φ).…”

Section: Counting Small Induced Subgraphsmentioning

confidence: 99%

Homomorphisms are a good basis for counting small subgraphs

Curticapean

Dell

Marx

2017

Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

182

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We introduce graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs H, the number of H-copies (induced or not) in an input graph G, and the number of homomorphisms from H to G.Using the framework of graph motif parameters, we obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G: For graphs H on k edges, we show how to count subgraph copies ofby a surprisingly simple algorithm. This improves upon previously known running times, such as O(n 0.91k+c ) time for k-edge matchings or O(n 0.46k+c ) time for k-cycles. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class C of such parameters, we consider the problem of evaluating f ∈ C on input graphs G, parameterized by the number of induced subgraphs that f depends upon. For every recursively enumerable class C, we prove the above problem to be either FPT or #W[1]-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way, together with improved lower bounds.Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy:

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“…where Φ k is the set of all (unlabeled) graphs with k vertices that satisfy Φ. The generality of #IndSub(Φ) allows to count almost arbitrary substructures in graphs, subsuming lots of parameterized counting problems that have been studied before, and hence the problem deserves a thorough complexity analysis with respect to the property Φ. Jerrum and Meeks proved it to be #W [1]-hard for the property of connectivity [14], for the property of having an even (or odd) number of edges [16] as well as for some other properties (see Section 1.2). As noted in [8], the theory of graph motif parameters immediately implies that for every property Φ, the problem #IndSub(Φ) is either fixed-parameter tractable or #W[1]-hard.…”

Section: Introductionmentioning

confidence: 99%

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

Roth

Schmitt

2020

Algorithmica

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We investigate the problem #IndSub(Φ) of counting all induced subgraphs of size k in a graph G that satisfy a given property Φ. This continues the work of Jerrum and Meeks who proved the problem to be #W[1]-hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even-or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties Φ, the problem #IndSub(Φ) is hard for #W[1] if the reduced Euler characteristic of the associated simplicial (graph) complex of Φ is nonzero. This observation links #IndSub(Φ) to Karp's famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the "topological approach to evasiveness" which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that #IndSub(Φ) is #W[1]-hard for every monotone property Φ that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not k-edge-connected for k > 2. Moreover, we show that for those properties #IndSub(Φ) can not be solved in time f (k) · n o(k) for any computable function f unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that #IndSub(Φ) is #W[1]-hard if Φ is any non-trivial modularity constraint on the number of edges with respect to some prime q or if Φ enforces the presence of a fixed isolated subgraph.

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The parameterised complexity of counting even and odd induced subgraphs

Cited by 17 publications

References 10 publications

Randomised Enumeration of Small Witnesses Using a Decision Oracle

Randomised Enumeration of Small Witnesses Using a Decision Oracle

Homomorphisms are a good basis for counting small subgraphs

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

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