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The Complexity of Approximately Counting Tree Homomorphisms

Leslie Ann Goldberg, Mark Jerrum
2014 ACM Transactions on Computation Theory  
We study two computational problems, parameterised by a fixed tree H. #HomsTo(H) is the problem of counting homomorphisms from an input graph G to H.  ...  There is an interesting connection between these homomorphism-counting problems and the problem of approximating the partition function of the ferromagnetic Potts model.  ...  In this paper, we show that the approximation problem is equivalent in complexity to a tree homomorphism problem.  ... 
doi:10.1145/2600917 fatcat:6n4fmor7irekvkfu66dri24m54

The Complexity of Approximately Counting Retractions [chapter]

Jacob Focke, Leslie Ann Goldberg, Stanislav Živný
2019 Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms  
The details of this, along with the generalisation of the framework to directed graphs, are given in Section 2.2.1 of the full version.  ...  It remains to show how to set up these instances to obtain the easiness result for partially bristled reflexive paths. This builds on the work of Kelk [31] .  ...  As we will see later, the complexity of approximately counting homomorphisms is still open (despite a lot of work on the problem) -even if restricted to trees H.  ... 
doi:10.1137/1.9781611975482.133 dblp:conf/soda/FockeGZ19 fatcat:5kbyvxtfr5erdf4ljei7ia4ary

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.  ...  the complexity of approximate counting, based largely on a connection with these phase transitions.  ... 
doi:10.4230/dagrep.7.8.23 dblp:journals/dagstuhl-reports/BezakovaGJ17 fatcat:yp3oqvgo4fal5lbio5yeqt4uje

Counting problems in parameterized complexity

Chihao Zhang, Yijia Chen
2014 Tsinghua Science and Technology  
The purpose of this article is to survey a few aspects of parameterized counting complexity, with a particular emphasis on some general frameworks in which parameterized complexity proves to be indispensable  ...  Parameterized complexity is a multivariate theory for the analysis of computational problems.  ...  We also discussed some randomized approximation counting algorithms. On the complexity side, we have a much refined view of counting problems.  ... 
doi:10.1109/tst.2014.6867521 fatcat:tluyfiannbgt3fmduir4ctsfje

Approximately Counting Answers to Conjunctive Queries with Disequalities and Negations [article]

Jacob Focke, Leslie Ann Goldberg, Marc Roth, Stanislav Živný
2022 arXiv   pre-print
We study the complexity of approximating the number of answers to a small query φ in a large database 𝒟.  ...  randomised Exponential Time Hypothesis (rETH) holds, then the problem has a fixed-parameter tractable approximation scheme (FPTRAS) if and only if the treewidth of φ is bounded. ∙ If the arity is unbounded  ...  one, the complexity of which was investigated by Dalmau and Jonsson [13] in the case of exact counting and by Bulatov and Živný [8] in the case of approximate counting.  ... 
arXiv:2103.12468v2 fatcat:qlf52fkbnvcatnhasbvbhyhj3e

Elastic Monte Carlo Tree Search with State Abstraction for Strategy Game Playing [article]

Linjie Xu, Jorge Hurtado-Grueso, Dominic Jeurissen, Diego Perez Liebana, Alexander Dockhorn
2022 arXiv   pre-print
To evaluate our method, we make use of the general strategy games platform Stratega to generate scenarios of varying complexity.  ...  In Elastic MCTS, the nodes of the tree are clustered dynamically, first grouped together progressively by state abstraction, and then separated when an iteration threshold is reached.  ...  At first, the tree grows as in normal MCTS (a). After a number of B iterations, ground nodes are grouped by using Approximate MDP Homomorphism (b).  ... 
arXiv:2205.15126v1 fatcat:e4ulu6bagzfhtlgay4pkdodzgm

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  ...  the exact complexity of the subgraph counting problem (STOC 17).  ...  Furthermore the author thanks Cornelius Brand for saying "Tutte Polynomial" every once in a while, and Philip Wellnitz for providing valuable feedback on early drafts of the full version of this work.  ... 
doi:10.1007/s00453-021-00805-y fatcat:oetq4yn2nnhy7ovyr47zgch6am

Counting Restricted Homomorphisms via Möbius Inversion over Matroid Lattices [article]

Marc Roth
2017 arXiv   pre-print
We use the general theorem to classify the complexity of counting locally injective homomorphisms as well as homomorphisms that are injective in the r-neighborhood for constant r.  ...  We present a framework for the complexity classification of parameterized counting problems that can be formulated as the summation over the numbers of homomorphisms from small pattern graphs H_1,...  ...  Furthermore the author thanks Cornelius Brand for saying "Tutte Polynomial" every once in a while.  ... 
arXiv:1706.08414v1 fatcat:54q2ljvnp5ewboak3d67ij73oq

Counting Homomorphisms to K4-minor-free Graphs, modulo 2 [chapter]

Jacob Focke, Leslie Ann Goldberg, Marc Roth, Stanislav Živný
2021 Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)  
As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3, as well as a full classification for the problem of counting list homomorphisms, modulo 2  ...  complexity class ⊕P of parity problems.  ...  Furthermore we thank Holger Dell for pointing out the tight conditional lower bound for counting independent sets modulo 2 in [5] .  ... 
doi:10.1137/1.9781611976465.137 fatcat:mdxd2toiofd4pmgy6dhibm7jwq

Approximate counting CSP seen from the other side [article]

Andrei A. Bulatov, Stanislav Zivny
2020 arXiv   pre-print
In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C,-), in which the goal is, given a relational structure A from a class C of structures and an  ...  arbitrary structure B, to find the number of homomorphisms from A to B.  ...  Acknowledgements We would like to thank the anonymous referees of both the conference [6] and this full version of the paper.  ... 
arXiv:1907.07922v2 fatcat:suaa4mv6czgv5pmhqupq2lr6ee

Counting Problems in Parameterized Complexity

Radu Curticapean, Michael Wagner
2019 International Symposium on Parameterized and Exact Computation  
After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large  ...  ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms, Theory of computation → Problems, reductions and completeness Keywords and phrases counting complexity,  ...  ] to classify the complexity of counting homomorphism variants such as locally injective homomorphisms.  ... 
doi:10.4230/lipics.ipec.2018.1 dblp:conf/iwpec/Curticapean18 fatcat:tdfs7zngzrgv5fixf2qawrccuy

FORESEE: Fully Outsourced secuRe gEnome Study basEd on homomorphic Encryption

Yuchen Zhang, Wenrui Dai, Xiaoqian Jiang, Hongkai Xiong, Shuang Wang
2015 BMC Medical Informatics and Decision Making  
., secure errorless division and secure approximation division) with a trade-off between complexity and accuracy in computing chi-square statistics.  ...  Remarkably, the secure approximation division provides significant performance gain, but without missing any significance SNPs in the chi-square association test using the aforementioned datasets.  ...  Complexity analysis in terms of cumulative circuit depth2 (CCD) and the number of homomorphic multiplications (HMs) for secure approximation division (Algorithm 3) integers.  ... 
doi:10.1186/1472-6947-15-s5-s5 pmid:26733391 pmcid:PMC4698942 fatcat:nf7klnm36fhmnl64rwps2x4ckq

The Complexity of Approximately Counting Retractions [article]

Jacob Focke, Leslie Ann Goldberg, Stanislav Zivny
2018 arXiv   pre-print
We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms  ...  Our second contribution is to locate the retraction counting problem in the complexity landscape of related approximate counting problems.  ...  As we will see later, the complexity of approximately counting homomorphisms is still open (despite a lot of work on the problem) -even if restricted to trees H.  ... 
arXiv:1807.00590v2 fatcat:vhsn3wqpebegnees66jgcwl34u

Approximate Counting CSP Seen from the Other Side

Andrei A. Bulatov, Stanislav Živný
2020 ACM Transactions on Computation Theory  
In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C, −), in which the goal is, given a relational structure A from a class C of structures and an  ...  arbitrary structure B, to find the number of homomorphisms from A to B.  ...  Only a handful of results exist for the approximation complexity of counting nonuniform CSPs. The approximation complexity of #CSP(−, {B}) for 2-element structures B was characterised by Dyer et al.  ... 
doi:10.1145/3389390 fatcat:isb3uf2umrachjbld6gie5el6y

Counting Subgraphs via Homomorphisms [chapter]

Omid Amini, Fedor V. Fomin, Saket Saurabh
2009 Lecture Notes in Computer Science  
We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms.  ...  polynomial, the classical algorithm of Kohn, Gottlieb, Kohn, and Karp for counting Hamiltonian cycles, Ryser's formula for counting perfect matchings of a bipartite graph, and color coding based algorithms  ...  Acknowledgement Many thanks to László Lovász for answering our questions on graph homomorphisms.  ... 
doi:10.1007/978-3-642-02927-1_8 fatcat:uw247ph62rbnjeggb2k3ysqq3y
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