Homomorphisms are a good basis for counting small subgraphs

Curticapean, Radu; Dell, Holger; Marx, Dániel

doi:10.1145/3055399.3055502

Cited by 79 publications

(182 citation statements)

References 62 publications

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“…While the proof Theorem 1.2 is conceptually close to the proof by Dalmau and Jonsson [19], we can combine Theorem 1.2 with the aforementioned complexity monotonicty [16] to lift the result to the realm of counting subgraphs: Theorem 1.3 (Intuitive version). Let F be a fixed graph and let H be a recursively enumerable class of graphs.…”

Section: Dichotomies For F -Colorable Graphs and König Graphsmentioning

confidence: 81%

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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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Section: Dichotomies For F -Colorable Graphs and König Graphsmentioning

confidence: 81%

“…Note that this construction induces self-loops if there is an edge between two vertices in the same block. Adopting the notation of [16], we denote quotient graphs without self-loops as spasms.…”

Section: Graphs and 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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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php input G to have loops, then a simpler interpolation based on a very recent paper by Curticapean, Dell and Marx [6] can be used to make the proofs very elegant! The exact same idea, written more generally, was also discovered by Chen [5].…”

Section: Related Workmentioning

confidence: 99%

“…A homomorphism from V (G) to V (H) is a compaction if it uses every vertex of H and also every non-loop edge of H (so it is surjective both on V (H) and on the non-loop edges in E(H)). Compactions have been studied under the name "homomorphic image" [18,22] and even under the name "surjective homomorphism" [6,24]. Once again, despite much work [1,27,28,29,30,31], there is still no characterisation of the complexity of determining whether there is a compaction from an input graph G to a graph H.…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Counting Surjective Homomorphisms and Compactions

Focke

Goldberg

Živný

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the vertices of H and all of the non-loop edges of H. Hell and Nešetřil gave a complete characterisation of the complexity of deciding whether there is a homomorphism from an input graph G to a fixed graph H. A complete characterisation is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a complete characterisation of the complexity of counting homomorphisms from an input graph G to a fixed graph H. In this paper, we give a complete characterisation of the complexity of counting surjective homomorphisms from an input graph G to a fixed graph H and we also give a complete characterisation of the complexity of counting compactions from an input graph G to a fixed graph H.

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“…We refer to e.g. Itai [72], and Curticapean, Dell, and Marx [40] for a non-exhaustive sample of earlier work on algorithm designs for subgraph counting. We postpone a discussion of hardness of the problem, implementations, and applications to §2.3.…”

mentioning

confidence: 99%

Engineering a Delegatable and Error-Tolerant Algorithm for Counting Small Subgraphs

Kaski

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

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We study the problem of counting the number of occurrences of a given six-vertex pattern graph S in an n-vertex host graph H. We engineer an open-source GPU implementation of a distributed algorithm design of Björklund and Kaski [PODC 2016] where (i) the execution of the algorithm can be delegated [Goldwasser, Kalai, and Rothblum, J. ACM 2015] to produce a noninteractive probabilistically checkable proof of correctness, and (ii) the execution of the algorithm when preparing the proof tolerates a controllable number of adversarial errors. Experiments with NVIDIA Tesla K80 and Tesla P100 Accelerators demonstrate that the framework is practical for inputs of up to 512 vertices, with proof checking being several orders of magnitude more efficient than preparing the proof; however, proof preparation still carries at least one order of magnitude overhead compared with just solving the problem.

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Homomorphisms are a good basis for counting small subgraphs

Cited by 79 publications

References 62 publications

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory

The Complexity of Counting Surjective Homomorphisms and Compactions

Engineering a Delegatable and Error-Tolerant Algorithm for Counting Small Subgraphs

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