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Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory [article]

Marc Roth, Philip Wellnitz
2021 arXiv   pre-print
Counting homomorphisms from a graph H into another graph G is a fundamental problem of (parameterized) counting complexity theory.  ...  counting homomorphisms from a graph in ℋ to a graph in 𝒢.  ...  Acknowledgements We thank Karl Bringmann and Holger Dell for fruitful discussions and valuable feedback on early drafts of this work.  ... 
arXiv:1907.03850v2 fatcat:rg3upoulmve7jlsnloszbm22zu

Counting problems in parameterized complexity

Chihao Zhang, Yijia Chen
2014 Tsinghua Science and Technology  
Parameterized complexity is a multivariate theory for the analysis of computational problems.  ...  Although the theory is mostly on decision problems, parameterized complexity naturally extends to counting problems as well.  ...  Conclusions In this short article, we surveyed various aspects of parameterized counting complexity, a theory combining parameterized (decision) complexity and classical counting complexity.  ... 
doi:10.1109/tst.2014.6867521 fatcat:tluyfiannbgt3fmduir4ctsfje

Counting Answers to Existential Questions

Holger Dell, Marc Roth, Philip Wellnitz, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
and Marx (FOCS 2014) for the subgraph counting problem.  ...  In particular, we are interested in the problem's parameterized and data complexity, where the query is considered to be small or even fixed, and the database is considered to be large.  ...  Acknowledgements We thank Cornelius Brand, Karl Bringmann, Radu Curticapean, Reinhard Diestel, Joshua Erde, Stephan Kreutzer, Stefan Mengel, Daniel Weißauer, and an anonymous reviewer for discussions and  ... 
doi:10.4230/lipics.icalp.2019.113 dblp:conf/icalp/DellRW19 fatcat:xfnlsag2bbefvnavyok634g46e

The complexity of counting homomorphisms seen from the other side

Víctor Dalmau, Peter Jonsson
2004 Theoretical Computer Science  
We prove (under a weaker complexity-theoretic assumption) that the corresponding counting problem #HOM(C, _) is solvable in polynomial time if and only if all structures in C have bounded tree-width.  ...  Grohe has proved that, under a certain complexity-theoretic assumption, HOM ( C, _) is solvable in polynomial time if and only if the cores of all structures in C have bounded tree-width.  ...  A minor map from A to B is a mapping M : A → B that is a minor map from G(A) to G(B). Parameterized complexity Finally, we need some facts concerning parameterized complexity theory.  ... 
doi:10.1016/j.tcs.2004.08.008 fatcat:gohfbujsq5egbosd62se2ae5ta

Understanding the Complexity of Induced Subgraph Isomorphisms [chapter]

Yijia Chen, Marc Thurley, Mark Weyer
2008 Lecture Notes in Computer Science  
For the homomorphism problem this kind of restriction has been studied by Grohe and Dalmau, Kolaitis and Vardi for the decision problem and by Dalmau and Jonsson for its counting variant.  ...  Under some assumption from parameterized complexity theory, these problems are solvable in polynomial time if and only if C contains no arbitrarily large graphs.  ...  Further thanks are due to Jörg Flum, Martin Grohe and Berit Grußien for comments on earlier versions of this paper.  ... 
doi:10.1007/978-3-540-70575-8_48 fatcat:vn45coq7i5a4rjwq3qb6fbunse

Counting Problems in Parameterized Complexity

Radu Curticapean, Michael Wagner
2019 International Symposium on Parameterized and Exact Computation  
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity.  ...  ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms, Theory of computation → Problems, reductions and completeness Keywords and phrases counting complexity,  ...  of these problems and paving the way for a rich complexity theory of parameterized counting problems.  ... 
doi:10.4230/lipics.ipec.2018.1 dblp:conf/iwpec/Curticapean18 fatcat:tdfs7zngzrgv5fixf2qawrccuy

The Parameterized Complexity of Counting Problems

Jörg Flum, Martin Grohe
2004 SIAM journal on computing (Print)  
We develop a parameterized complexity theory for counting problems.  ...  As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W[t], for t ≥ 1, that corresponds to Downey and Fellows's W-hierarchy [13] and show that a few central  ...  In this paper, we develop a basic parameterized complexity theory for counting problems.  ... 
doi:10.1137/s0097539703427203 fatcat:auo3obkzebf5rd4ruyxriwdheq

Counting Answers to Existential Questions [article]

Holger Dell, Marc Roth, Philip Wellnitz
2019 arXiv   pre-print
In particular, we are interested in the problem's parameterized and data complexity, where the query is considered to be small or fixed, and the database is considered to be large.  ...  (FOCS 2014) for the subgraph counting problem.  ...  In particular, we are very grateful to Joshua and Daniel for pointing out the reference [31] .  ... 
arXiv:1902.04960v2 fatcat:sm53qhw6dzanleeutgyztfxqdu

Parameterized Counting of Partially Injective Homomorphisms

Marc Roth
2021 Algorithmica  
AbstractWe study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting  ...  of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities.  ...  Acknowledgements The author is very grateful to Holger Dell and Radu Curticapean for fruitful discussions, as well as to an anonymous reviewer for pointing out the applicability of the framework to equi-join  ... 
doi:10.1007/s00453-021-00805-y fatcat:oetq4yn2nnhy7ovyr47zgch6am

The Fine Classification of Conjunctive Queries and Parameterized Logarithmic Space

Hubie Chen, Moritz Müller
2015 ACM Transactions on Computation Theory  
Our work yields a significantly richer perspective on the complexity of conjunctive queries and, at the same time, suggests new avenues of research in parameterized complexity.  ...  We perform a fundamental investigation of the complexity of conjunctive query evaluation from the perspective of parameterized complexity.  ...  Another avenue for future research is to develop the theory of the degrees of counting problems identified by our counting classification.  ... 
doi:10.1145/2751316 fatcat:4vkqscdtsvd7jm7uyqd7237rqa

The Fine Classification of Conjunctive Queries and Parameterized Logarithmic Space Complexity [article]

Hubie Chen, Moritz Müller
2013 arXiv   pre-print
Our work yields a significantly richer perspective on the complexity of conjunctive queries and, at the same time, suggests new avenues of research in parameterized complexity.  ...  We perform a fundamental investigation of the complexity of conjunctive query evaluation from the perspective of parameterized complexity.  ...  Acknowledgements The first author was supported by the Spanish Project FORMALISM (TIN2007-66523), by the Basque Government Project S-PE12UN050(SAI12/219), and by the University of the Basque Country under  ... 
arXiv:1306.5424v2 fatcat:qvh3vgimjjcs7fth2mlg6hrdhi

The fine classification of conjunctive queries and parameterized logarithmic space complexity

Hubie Chen, Moritz Müller
2013 Proceedings of the 32nd symposium on Principles of database systems - PODS '13  
Our work yields a significantly richer perspective on the complexity of conjunctive queries and, at the same time, suggests new avenues of research in parameterized  ...  We perform a fundamental investigation of the complexity of conjunctive query evaluation from the perspective of parameterized complexity.  ...  Acknowledgements The first author was supported by the Spanish Project FORMALISM (TIN2007-66523), by the Basque Government Project S-PE12UN050(SAI12/219), and by the University of the Basque Country under  ... 
doi:10.1145/2463664.2463669 dblp:conf/pods/ChenM13 fatcat:vwwa6gk5ujbkvhky73qcbfkb3u

The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 15301)

Andrei A. Bulatov, Venkatesan Guruswami, Andrei Krokhin, Dániel Marx, Marc Herbstritt
2016 Dagstuhl Reports  
During the past two decades, an impressive array of diverse methods from several different mathematical fields, including algebra, logic, mathematical programming, probability theory, graph theory, and  ...  The Dagstuhl Seminar 15301 "The Constraint Satisfaction Problem: Complexity and Approximability" was aimed at bringing together researchers using all the different techniques in the study of the CSP, so  ...  ), model checking existential positive formulas (Chen '14), and counting homomorphisms (Dalmau and Jonsson '04).  ... 
doi:10.4230/dagrep.5.7.22 dblp:journals/dagstuhl-reports/BulatovGKM15 fatcat:lcyvfsxsbzbrfe27irlpxepxoi

Computational Counting (Dagstuhl Seminar 18341)

Ivona Bezáková, Leslie Ann Goldberg, Mark R. Jerrum, Marc Herbstritt
2018 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 17341 "Computational Counting".  ...  A total of 43 researchers from all over the world, with interests and expertise in different aspects of computational counting, actively participated in the meeting.  ...  We provide a complete parameterized complexity dichotomy for counting such homomorphisms, that is, we identify classes of patterns for which the problem is fixed-parameter tractable (FPT), including an  ... 
doi:10.4230/dagrep.7.8.23 dblp:journals/dagstuhl-reports/BezakovaGJ17 fatcat:yp3oqvgo4fal5lbio5yeqt4uje

Limits and Applications of Group Algebras for Parameterized Problems [chapter]

Ioannis Koutis, Ryan Williams
2009 Lecture Notes in Computer Science  
Among other results, we show that a combination of dynamic programming and a variation of the algebraic method can break the trivial upper bounds for exact parameterized counting in fairly general settings  ...  In this paper we use communication complexity to show that the O * (2 k ) algorithm is essentially optimal within this evaluation oracle framework.  ...  One application of this is an O * (n mk/2 ) algorithm for counting k-packings of m-sets over a universe of size n.  ... 
doi:10.1007/978-3-642-02927-1_54 fatcat:v2yxdzzilrautoovjprqdivd2e
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