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A complexity trichotomy for approximately counting list H-colourings [article]

Andreas Galanis, Leslie Ann Goldberg, Mark Jerrum
2017 arXiv   pre-print
We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H.  ...  For every other graph H, approximately counting list H-colourings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP=RP).  ...  Overview In this paper we study the complexity of approximately counting the list H-colourings of a graph.  ... 
arXiv:1602.03985v6 fatcat:aadksachnnhdtihegszc6tsbyy

A Complexity Trichotomy for Approximately Counting List H-Colourings * † ‡ §

Andreas Galanis, Leslie Goldberg, Mark Jerrum
unpublished
We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H.  ...  For every other graph H, approximately counting list H-colourings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP = RP).  ...  This holds for every edge, so σ is an H-colouring of G. I C A L P 2 0 1 6 46:12 A Complexity Trichotomy for Approximately Counting List H-Colourings Finally, we add clauses to deal with lists.  ... 
fatcat:ww6ua37kijgtjirdruyw6uynl4

The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)

Martin Grohe, Venkatesan Guruswami, Stanislav Zivny, Michael Wagner
2018 Dagstuhl Reports  
Constraint satisfaction has always played a central role in computational complexity theory; appropriate versions of CSPs are classical complete problems for most standard complexity classes.  ...  For instance, they help to understand which mathematical properties make a computational problem tractable (in a wide sense, e.g., polynomial-time solvable, non-trivially approximable, fixed-parameter  ...  -The Constraint Satisfaction Problem: Complexity and Approximability From Weak to Strong LP Gaps for all CSPs We study the approximability of constraint satisfaction problems (CSPs) by linear programming  ... 
doi:10.4230/dagrep.8.6.1 dblp:journals/dagstuhl-reports/GroheGZ18 fatcat:3bqo62ly3rgzlnh3bmkvwbuwea

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  
Let G be a graph that contains an induced subgraph H.  ...  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] .  ...  We then show that the problem #Ret(H I v ,I e ) reduces to the #BIS-easy problem #CSP({Imp, δ 0 , δ 1 }). By choosing different instances I v and I e one can generate different #BIS-easiness results.  ... 
doi:10.1137/1.9781611975482.133 dblp:conf/soda/FockeGZ19 fatcat:5kbyvxtfr5erdf4ljei7ia4ary

The Complexity of Approximately Counting Tree Homomorphisms

Leslie Ann Goldberg, Mark Jerrum
2014 ACM Transactions on Computation Theory  
Even though H is a tree, these problems turn out to be sufficiently rich to capture all of the known approximation behaviour in #P. We give a complete trichotomy for #WHomsTo(H).  ...  We use this connection to obtain some additional upper bounds for the approximation complexity of #HomsTo(J_q).  ...  For this problem, we show that there is a complexity trichotomy, and the trichotomy depends upon the induced subgraphs of H.  ... 
doi:10.1145/2600917 fatcat:6n4fmor7irekvkfu66dri24m54

Approximately counting and sampling small witnesses using a colourful decision oracle [article]

Holger Dell, John Lapinskas, Kitty Meeks
2019 arXiv   pre-print
From an algorithmic standpoint, our results will make the development of new approximate counting algorithms substantially easier; indeed, it already yields a new state-of-the-art algorithm for approximately  ...  In this paper, we prove "black box" results for turning algorithms which decide whether or not a witness exists into algorithms to approximately count the number of witnesses, or to sample from the set  ...  Nevertheless, we are able to use Theorem 1.7 to give a fine-grained reduction from approximate #Colourful-H to Colourful-H and from approximate #Weighted-H to Weighted-H for all graphs H; see Corollary  ... 
arXiv:1907.04826v1 fatcat:66xedkrptvhotp3hqiid2b4wmq

The Complexity of Approximating Bounded-Degree Boolean #CSP (Extended Abstract) [article]

Martin E. Dyer, Leslie Ann Goldberg, Markus Jalsenius, David Richerby
2010 arXiv   pre-print
For lower degree bounds, additional cases arise in which the complexity is related to the complexity of approximately counting independent sets in hypergraphs.  ...  We consider the approximate counting problem for Boolean CSPs with bounded-degree instances, for constraint languages containing the two unary constant relations 0 and 1.  ...  Our main result, Corollary 6.6, is a trichotomy for the case in which instances have maximum degree d for some d 25.  ... 
arXiv:1001.4987v2 fatcat:4tpnzyirivbjfglxnj7koxf2ky

The Complexity of Approximating Bounded-Degree Boolean CSP [article]

Martin Dyer, Leslie Ann Goldberg, Markus Jalsenius, David Richerby
2011 arXiv   pre-print
For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs.  ...  We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations 0 and 1.  ...  Counting CSP result follows immediately from Courcelle's theorem, which says that, if a decision problem is definable in monadic second-order logic (which H-colouring is, for any fixed H), then both it  ... 
arXiv:0907.2663v2 fatcat:wywiy6ybmnfcflztiuajewnrsi

On emergence in gauge theories at the 't Hooft limit

Nazim Bouatta, Jeremy Butterfield
2014 European Journal for Philosophy of Science  
The idea of the limit is that the number N of colours (or charges) goes to infinity.  ...  These properties are planarity and integrability, as in (i) and (ii) above; and the behaviour of the beta-function reflecting, for example, asymptotic freedom.  ...  Acknowledgements:-This work was supported by a grant from Templeton World Charity Foundation.  ... 
doi:10.1007/s13194-014-0098-1 fatcat:qj2tz2jxtvhspgz2vss246bdpq

The complexity of approximating bounded-degree Boolean #CSP

Martin Dyer, Leslie Ann Goldberg, Markus Jalsenius, David Richerby
2012 Information and Computation  
For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs.  ...  We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}.  ...  counting H -colouring problem.  ... 
doi:10.1016/j.ic.2011.12.007 fatcat:hiclseccibgshoiem5tsgkecsq

The Complexity of Approximately Counting Retractions to Square-Free Graphs [article]

Jacob Focke, Leslie Ann Goldberg, Stanislav Živný
2021 arXiv   pre-print
We give a complete trichotomy for the complexity of approximately counting retractions to all square-free graphs (graphs that do not contain a cycle of length 4).  ...  By giving new #BIS-easiness results we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs which were previously unresolved.  ...  In this work we give a complete complexity trichotomy for approximately counting retractions to all square-free graphs (graphs that do not contain a 4-cycle) and we show that all of these problems fall  ... 
arXiv:1907.02319v4 fatcat:lrrzh6ya5vbfrp6txegcae26sy

On emergence in gauge theories at the 't Hooft limit [article]

Nazim Bouatta, Jeremy Butterfield
2012 arXiv   pre-print
The idea of the limit is that the number N of colours (or charges) goes to infinity.  ...  These properties are planarity and integrability, as in (i) and (ii) above; and the behaviour of the beta-function reflecting, for example, asymptotic freedom.  ...  For a sphere and its homeo- morphs: h = 0, b = 0, and so χ = 2; for a torus and its homeomorphs, h = 1, b = 0, and so χ = 0.  ... 
arXiv:1208.4986v1 fatcat:4wl3cmhdsvf5ljbsrfdy7z7a34

From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook [article]

Yang-Hui He
2022 arXiv   pre-print
With this paradigm as a model for human intuition - complementary to and in contrast with the more formalistic approach of automated theorem proving - we highlight some experiments on how AI helps with  ...  The author is grateful to STFC UK for grant ST/J00037X/2, as well as the kind hospitality -virtual and in person (!)  ...  during 2021 -of Cambridge (Winter School on ML), Cairo (BSM 22), Sofia (Lie Theory in Physics XIV), Toulouse (Geometry, Topology & AI), Lisboa (BH, BPS, & QI), Trento (ML for HEP), Singapore (Mtheory &  ... 
arXiv:2202.06086v1 fatcat:rhg2tovnpfactcdq6t4d42msre

The Complexity of Approximately Counting Retractions [article]

Jacob Focke, Leslie Ann Goldberg, Stanislav Zivny
2018 arXiv   pre-print
Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs of girth at least 5.  ...  Retractions are very well-studied: Given H, the complexity of deciding whether there is a retraction from an input graph G to H is completely classified, in the sense that it is known for which H this  ...  The complexity of approximately counting list homomorphisms is known, due to Galanis, Goldberg and Jerrum [16] .  ... 
arXiv:1807.00590v2 fatcat:vhsn3wqpebegnees66jgcwl34u

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  ...  that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C,-).  ...  The complexity of CSP(−, {H}), for a fixed graph H, was studied under the name of H-colouring by Hell and Nešetřil [34] .  ... 
arXiv:1907.07922v2 fatcat:suaa4mv6czgv5pmhqupq2lr6ee
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