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

2017
*
arXiv
*
pre-print

We examine the computational

arXiv:1602.03985v6
fatcat:aadksachnnhdtihegszc6tsbyy
*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. ...##
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A Complexity Trichotomy for Approximately Counting List H-Colourings * † ‡ §

unpublished

We examine the computational

fatcat:ww6ua37kijgtjirdruyw6uynl4
*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*. ...##
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The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)

2018
*
Dagstuhl Reports
*

Constraint satisfaction has always played

doi:10.4230/dagrep.8.6.1
dblp:journals/dagstuhl-reports/GroheGZ18
fatcat:3bqo62ly3rgzlnh3bmkvwbuwea
*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 ...##
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The Complexity of Approximately Counting Retractions
[chapter]

2019
*
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms
*

Let G be

doi:10.1137/1.9781611975482.133
dblp:conf/soda/FockeGZ19
fatcat:5kbyvxtfr5erdf4ljei7ia4ary
*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. ...##
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The Complexity of Approximately Counting Tree Homomorphisms

2014
*
ACM Transactions on Computation Theory
*

Even though

doi:10.1145/2600917
fatcat:6n4fmor7irekvkfu66dri24m54
*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*. ...##
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Approximately counting and sampling small witnesses using a colourful decision oracle
[article]

2019
*
arXiv
*
pre-print

From an algorithmic standpoint, our results will make the development of new

arXiv:1907.04826v1
fatcat:66xedkrptvhotp3hqiid2b4wmq
*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 ...##
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The Complexity of Approximating Bounded-Degree Boolean #CSP (Extended Abstract)
[article]

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. ...

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

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 ...

##
###
On emergence in gauge theories at the 't Hooft limit

2014
*
European Journal for Philosophy of Science
*

The idea of the limit is that the number N of

doi:10.1007/s13194-014-0098-1
fatcat:qj2tz2jxtvhspgz2vss246bdpq
*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. ...##
###
The complexity of approximating bounded-degree Boolean #CSP

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. ...

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

2021
*
arXiv
*
pre-print

We give

arXiv:1907.02319v4
fatcat:lrrzh6ya5vbfrp6txegcae26sy
*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 ...##
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On emergence in gauge theories at the 't Hooft limit
[article]

2012
*
arXiv
*
pre-print

The idea of the limit is that the number N of

arXiv:1208.4986v1
fatcat:4wl3cmhdsvf5ljbsrfdy7z7a34
*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. ...##
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From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook
[article]

2022
*
arXiv
*
pre-print

With this paradigm as

arXiv:2202.06086v1
fatcat:rhg2tovnpfactcdq6t4d42msre
*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 & ...##
###
The Complexity of Approximately Counting Retractions
[article]

2018
*
arXiv
*
pre-print

Our first contribution is to give

arXiv:1807.00590v2
fatcat:vhsn3wqpebegnees66jgcwl34u
*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] . ...##
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Approximate counting CSP seen from the other side
[article]

2020
*
arXiv
*
pre-print

In this paper we study the

arXiv:1907.07922v2
fatcat:suaa4mv6czgv5pmhqupq2lr6ee
*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] . ...
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