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Homomorphisms are a good basis for counting small subgraphs

Radu Curticapean, Holger Dell, Dániel Marx
2017 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017  
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.  ...  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  ...  Acknowledgments Thanks a lot to Édouard Bonnet for pointing out [52] and [22] .  ... 
doi:10.1145/3055399.3055502 dblp:conf/stoc/CurticapeanDM17 fatcat:elq7lwyoyffbzbdiwlxycit2jq

Counting Subgraphs in Degenerate Graphs [article]

Suman K. Bera, Lior Gishboliner, Yevgeny Levanzov, C. Seshadhri, Asaf Shapira
2021 arXiv   pre-print
Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph H to be easy.  ...  We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms.  ...  Marx, Homomorphisms are a good basis for counting small subgraphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 210- 223, 2017. [20] R.  ... 
arXiv:2010.05998v2 fatcat:uzcii64jinfulh2tao7sq7iqau

Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles [article]

Suman K. Bera, Noujan Pashanasangi, C. Seshadhri
2020 arXiv   pre-print
We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting.  ...  Counting homomorphisms of a constant sized pattern graph H in an input graph G is a fundamental computational problem.  ...  While we discover a clean dichotomy for the homomorphism counting problem, the landscape for the subgraph counting problem is not as clear.  ... 
arXiv:2010.08083v2 fatcat:oryutbr3ljc3hnyebu5mmjgtha

Subgraph Frequencies: Mapping the Empirical and Extremal Geography of Large Graph Collections [article]

Johan Ugander, Lars Backstrom, Jon Kleinberg
2013 arXiv   pre-print
In this work, we draw on the theory of graph homomorphisms to formulate and analyze such a representation, based on computing the frequencies of small induced subgraphs within each graph.  ...  For the extremal properties, we develop a tractable linear program for bounding the feasible space of subgraph frequencies by harnessing a toolkit of known extremal graph theory.  ...  We thank Peter Grassberger for helpful comments. This work was supported in part by NSF grants IIS-0910664 and IIS-1016099.  ... 
arXiv:1304.1548v2 fatcat:entafhr45nhkvp7dtup6gjh2iy

The Glasgow Subgraph Solver: Using Constraint Programming to Tackle Hard Subgraph Isomorphism Problem Variants [chapter]

Ciaran McCreesh, Patrick Prosser, James Trimble
2020 Lecture Notes in Computer Science  
It combines constraint programming concepts with a variety of strong but fast domain-specific search and inference techniques, and is suitable for use on a wide range of graphs, including many that are  ...  The Glasgow Subgraph Solver provides an implementation of state of the art algorithms for subgraph isomorphism problems.  ...  We would like to thank Blair Archibald, Fraser Dunlop, Jan Elffers, Stephan Gocht, Ruth Hoffmann, Jakob Nordström, and Christine Solnon for their contributions to the design and implementation of the solver  ... 
doi:10.1007/978-3-030-51372-6_19 fatcat:orm2rsy5obabnfkb34fghx7gv4

Subgraph frequencies

Johan Ugander, Lars Backstrom, Jon Kleinberg
2013 Proceedings of the 22nd international conference on World Wide Web - WWW '13  
In this work, we draw on the theory of graph homomorphisms to formulate and analyze such a representation, based on computing the frequencies of small induced subgraphs within each graph.  ...  For the extremal properties, we develop a tractable linear program for bounding the feasible space of subgraph frequencies by harnessing a toolkit of known extremal graph theory.  ...  More precisely, we choose a small number k (e.g. k = 3 or 4); then, for each graph G in a collection, we create a vector with a coordinate for each distinct k-node subgraph H, specifying the fraction of  ... 
doi:10.1145/2488388.2488502 dblp:conf/www/UganderBK13 fatcat:rjtmmwlqdbac7e23vh2xv3g6hm

The challenges of unbounded treewidth in parameterised subgraph counting problems

Kitty Meeks
2016 Discrete Applied Mathematics  
Parameterised subgraph counting problems are the most thoroughly studied topic in the theory of parameterised counting, and there has been significant recent progress in this area.  ...  ; here, we prove a number of hardness results for the situation in which this bounded treewidth condition does not hold, resulting in dichotomies for some special cases of the general subgraph counting  ...  Acknowledgement I am very grateful to Mark Jerrum for his helpful comments on several previous drafts of this paper.  ... 
doi:10.1016/j.dam.2015.06.019 fatcat:utjeznnsxfd45oozdipbxe5xyu

The challenges of unbounded treewidth in parameterised subgraph counting problems [article]

Kitty Meeks
2015 arXiv   pre-print
; here, we prove a number of hardness results for the situation in which this bounded treewidth condition does not hold, resulting in dichotomies for some special cases of the general subgraph counting  ...  Parameterised subgraph counting problems are the most thoroughly studied topic in the theory of parameterised counting, and there has been significant recent progress in this area.  ...  Acknowledgements I am very grateful to Mark Jerrum for his helpful comments on several previous drafts of this paper. References  ... 
arXiv:1402.5857v2 fatcat:e2qzddfidbe73airuxbpmi4lwm

Regularity inheritance in hypergraphs [article]

Peter Allen, Ewan Davies, Jozef Skokan
2019 arXiv   pre-print
We give a new approach to handling hypergraph regularity.  ...  We also prove a corresponding sparse hypergraph regularity lemma.  ...  Given a bounded-degree complex H, Theorem 3 gives sufficient counting conditions for a good lower bound on the number of homomorphisms from H into Γ.  ... 
arXiv:1901.05955v1 fatcat:7yod52yeuzdpxpgbdyixawrzbm

Regularity method and large deviation principles for the Erdős–Rényi hypergraph [article]

Nicholas A. Cook, Amir Dembo, Huy Tuan Pham
2021 arXiv   pre-print
Combined with associated counting lemmas, these yield sharp asymptotics for upper tails of homomorphism counts in the r-uniform Erdős–Rényi hypergraph for any fixed r≥ 2, generalizing and improving on  ...  We develop a quantitative large deviations theory for random Bernoulli tensors.  ...  Joint upper tails for multiple subgraph counts.  ... 
arXiv:2102.09100v2 fatcat:2e7azmtbgzcpfk3hojnqqlt5di

APPLICATIONS OF GRAPH PROBING TO WEB DOCUMENT ANALYSIS [chapter]

Daniel Lopresti, Gordon Wilfong
2003 Series in Machine Perception and Artificial Intelligence  
This observation can be seen as forming the basis for graph probing, a paradigm we have recently begun exploring for graph comparison [5, 8] .  ...  Valiente and Martínez describe an approach for subgraph pattern-matching based on finding homomorphic images of every connected component in the query [12] .  ... 
doi:10.1142/9789812775375_0002 fatcat:jrtmng2ixnh7bjl5dxcfili7li

Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders [article]

Marc Roth, Johannes Schmitt, Philip Wellnitz
2021 arXiv   pre-print
As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of #𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ).  ...  Given a graph property Φ, we consider the problem 𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ), where the input is a pair of a graph G and a positive integer k, and the task is to decide whether G contains a k-edge subgraph that  ...  The second author was supported by the SNF early postdoc mobility grant 184245 and thanks the Max Planck Institute for Mathematics in Bonn for its hospitality.  ... 
arXiv:2011.03433v2 fatcat:7aauf7xejvfx5atw3n7s6sefsa

Chromatic number, induced cycles, and non-separating cycles [article]

Hanbaek Lyu
2018 arXiv   pre-print
As an application, we show that a 3-connected graph with average degree k> 2 have at least (k-1)|V|+Ck^3^3/2k induced non-separating cycles for some explicit constant C>0.  ...  We also give a short proof of this theorem of Tutte.  ...  But sometimes, simply counting the number of basic objects such as vertices, edges, and cycles, gives a good understanding on some structural properties of the graphs.  ... 
arXiv:1211.0627v7 fatcat:axweu6n5gzhm7fmll5gwph66i4

Graph sampling with applications to estimating the number of pattern embeddings and the parameters of a statistical relational model

Irma Ravkic, Martin Žnidaršič, Jan Ramon, Jesse Davis
2018 Data mining and knowledge discovery  
In contrast to earlier work, which focused on exact counting for simple (i.e., very short) patterns, we present a sampling approach for estimating the statistics of larger graph pattern occurrences.  ...  Abstract Counting the number of times a pattern occurs in a database is a fundamental data mining problem.  ...  Graph particles are usually small subgraphs that contain up to five nodes, and are sometimes called graphlets.  ... 
doi:10.1007/s10618-018-0553-2 fatcat:mlpacwxqpzdhjk5btkpx7d26ym

Counting Motifs with Graph Sampling [article]

Jason M. Klusowski, Yihong Wu
2018 arXiv   pre-print
This is shown to be optimal for all motifs with at most 4 vertices and cliques of all sizes. The matching minimax lower bounds are established using certain algebraic properties of subgraph counts.  ...  We show that: for any connected h on k vertices, to estimate s=s(h,G), the number of copies of h in the parent graph G of maximum degree d, with a multiplicative error of ϵ, (a) For subgraph sampling,  ...  Nonetheless, we can still find a good estimate.  ... 
arXiv:1802.07773v1 fatcat:2wldd24eyvajdbk772eogjqm4u
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