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Counting independent sets and colorings on random regular bipartite graphs
[article]

2019
*
arXiv
*
pre-print

We give a fully polynomial-time approximation scheme (FPTAS) to

arXiv:1903.07531v1
fatcat:enbkxxpa2vdz7p754dwz235zjq
*count*the number of*independent**sets**on*almost every Δ-*regular**bipartite**graph*if Δ> 53. ... , there is an FPTAS to*count*the number of q-*colorings**on*almost every Δ-*regular**bipartite**graph*. ...*Random**regular**bipartite**graphs*frequently appear in the analysis of hardness of*counting**independent**sets*[MWW09, DFJ02, Sly10, SS12, GŠVY16] . ...##
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Counting Independent Sets and Colorings on Random Regular Bipartite Graphs

2019
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

We give a fully polynomial-time approximation scheme (FPTAS) to

doi:10.4230/lipics.approx-random.2019.34
dblp:conf/approx/LiaoLLM19
fatcat:ip6tpov4sjbb3a5k54pxpvzlbm
*count*the number of*independent**sets**on*almost every ∆-*regular**bipartite**graph*if ∆ ≥ 53. ... (q), there is an FPTAS to*count*the number of q-*colorings**on*almost every ∆-*regular**bipartite**graph*. ... We follow this 34:4*Counting**Independent**Sets**and**Colorings**on**Random**Regular**Bipartite**Graphs*idea*and*define a polymer model representing deviations from a ground cluster. ...##
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Algorithms for #BIS-hard problems on expander graphs
[chapter]

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

The class #BIS is the class of problems polynomial-time equivalent to approximating the number of

doi:10.1137/1.9781611975482.135
dblp:conf/soda/JenssenKP19
fatcat:nvvcwyd7d5hj3ashc27iphf4aa
*independent**sets*of a*bipartite**graph*[10],*and*many interesting approximate*counting**and*sampling problems ... For #BIS, we would like to find subclasses of*bipartite**graphs**on*which we can efficiently approximate the number of*independent**sets*or the hard-core partition function. ... This would furnish us with an FPTAS for*counting**independent**sets**and*sampling*independent**sets*uniformly in the*random**regular**bipartite**graph*. ...##
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Algorithms for #BIS-Hard Problems on Expander Graphs

2020
*
SIAM journal on computing (Print)
*

This would furnish us with an FPTAS for

doi:10.1137/19m1286669
fatcat:cvn3rdusmncibiansjubwpz434
*counting**independent**sets**and*sampling*independent**sets*uniformly in the*random**regular**bipartite**graph*. ... We state a conjecture in this direction*on*proper*colorings*in*random**regular**bipartite**graphs*, but*one*could also make conjectures along the same lines for all*bipartite*expander*graphs*or for the anti-ferromagnetic ...##
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Counting Colorings of a Regular Graph

2014
*
Graphs and Combinatorics
*

• The

doi:10.1007/s00373-013-1403-z
fatcat:qxets5cmk5cp5ijzvn3u66qjme
*independent**set*sequence of*regular**bipartite**graphs*, Discrete Math. 312 (2012), 2881- 2892, arXiv:1110.3760 • H-colouring*bipartite**graphs*(with J. ... B 102 An upper bound for the number of*independent**sets*in*regular**graphs*, Discrete Math. Matchings*and**Independent**Sets*of a Fixed Size in*Regular**Graphs*(with T. Carroll*and*P. ...##
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Sidorenko's conjecture, colorings and independent sets
[article]

2017
*
arXiv
*
pre-print

These cases correspond to

arXiv:1603.05888v4
fatcat:a5xccocz2jhczdd2e4mgtnlw54
*counting**colorings*,*independent**sets**and*Widom-Rowlinson*colorings*of a*graph*H. ... In fact, we will prove that in the last two cases (*independent**sets**and*Widom-Rowlinson*colorings*) the*graph*H does not need to be*bipartite*. ... d) is the*set*of d-*regular**graphs**on*n vertices with girth at least g. ...##
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Sidorenko's Conjecture, Colorings and Independent Sets

2017
*
Electronic Journal of Combinatorics
*

These cases correspond to

doi:10.37236/6019
fatcat:vlfw7vy6kzg6vkbaztagwidgm4
*counting**colorings*,*independent**sets**and*Widom-Rowlinson*colorings*of a*graph*$H$. ... $$In fact, we will prove that in the last two cases (*independent**sets**and*Widom-Rowlinson*colorings*) the*graph*$H$ does not need to be*bipartite*. ... DMS-1500219,*and*by the MTA Rényi "Lendüle" Groups*and**Graphs*Research Group,*and*by the ERC Consolidator Grant 648017,*and*by the Hungarian National Research, Development*and*Innovation Office, NKFIH ...##
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Algorithms for #BIS-hard problems on expander graphs
[article]

2020
*
arXiv
*
pre-print

We also find efficient

arXiv:1807.04804v3
fatcat:bhldiwazznazpokcqnclo6xrde
*counting**and*sampling algorithms for proper q-*colorings*of*random*Δ-*regular**bipartite**graphs*when q is sufficiently small as a function of Δ. ... We give an FPTAS*and*an efficient sampling algorithm for the high-fugacity hard-core model*on*bounded-degree*bipartite*expander*graphs**and*the low-temperature ferromagnetic Potts model*on*bounded-degree ... In particular, by*setting*λ = 1 this gives an FPTAS for*counting*the total number of*independent**sets*in*random*∆-*regular**bipartite**graphs*for large enough ∆. Remark 1. ...##
###
Three tutorial lectures on entropy and counting
[article]

2014
*
arXiv
*
pre-print

We explain the notion of the entropy of a discrete

arXiv:1406.7872v1
fatcat:cvqnryzj5jfnphyooktpqmhcae
*random*variable,*and*derive some of its basic properties. ... bound*on*the number of*independent**sets*admitted by a*regular**bipartite**graph*. • Kahn, Range of cube-indexed*random*walk (2001) [36] : Answers a question of Benjamini, Häggström*and*Mossel*on*the typical ... A natural question to ask is what happens to the bounds*on*number of q-*colorings**and*homomorphism*counts*, when we relax to the family of general (not necessarily*bipartite*) n-vertex, d-*regular**graphs*; ...##
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On Markov Chains for Randomly H-Coloring a Graph

2001
*
Journal of Algorithms
*

The probabilistic model we consider includes

doi:10.1006/jagm.2000.1142
fatcat:6wg4j2ekvngqtjqrlkpgsfc6fy
*random*proper*colorings*,*random**independent**sets*,*and*the Widom-Rowlinson*and*Beach models of statistical physics. ... We will often refer to c v as the*color*of v*and*c as an H-*coloring*of G. We consider the problem of choosing a*random*H-*coloring*of G by Markov chain Monte Carlo. ... In all cases, the "bad"*graphs*we use are either*random*r-*regular**graphs*or r-*regular**bipartite**graphs*for sufficiently large (but bounded) r. ...##
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1-factorizations of pseudorandom graphs
[article]

2018
*
arXiv
*
pre-print

Specifically, we prove that an (n,d,λ)-

arXiv:1803.10361v1
fatcat:src4qm5kazd6pixztnwh2bzczm
*graph*G (that is, a d-*regular**graph**on*n vertices whose second largest eigenvalue in absolute value is at most λ) admits a 1-factorization provided that n is even ... all possible values of d results obtained by Janson,*and**independently*by Molloy, Robalewska, Robinson,*and*Wormald for fixed d. ... A*graph*D*on*the vertex*set*[n] is called a dependency*graph*for (A i ) i if A i is mutually*independent*of all the events {A j : ij / ∈ E(D)}. ...##
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Page 4037 of Mathematical Reviews Vol. , Issue 2003f
[page]

2003
*
Mathematical Reviews
*

*independent*

*set*, (v) a linear-time algorithm for finding a minimum dominating

*set*, (vi) an O(\/nm) algorithm for finding an optimal

*coloring*. ... 4037 edge-

*color*a

*bipartite*

*graph*with n nodes, m edges,

*and*maximum degree A.” 2003f:05098 05C75 Hempel, Harald (D-FSUMI,; Jena); Kratsch, Dieter (D-FSUMI; Jena)

*On*claw-free asteroidal triple-free

*graphs*...

##
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Counting dominating sets and related structures in graphs
[article]

2015
*
arXiv
*
pre-print

We consider some problems concerning the maximum number of (strong) dominating

arXiv:1503.00998v1
fatcat:zrxingtkwfadlnqgnu5miffgxe
*sets*in a*regular**graph*,*and*their weighted analogues. Our primary tool is Shearer's entropy lemma. ... These techniques extend to a reasonably broad class of*graph*parameters enumerating vertex*colorings*satisfying conditions*on*the multiset of*colors*appearing in (closed) neighborhoods. ... If G is an r-*regular**bipartite**graph**on*n vertices*and*H is any*graph*(which may have loops), then hom(G, H) ≤ hom(K r,r , H) n/(2r) . ...##
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Approximately counting independent sets in bipartite graphs via graph containers
[article]

2021
*
arXiv
*
pre-print

Finally we present an algorithm that applies to all d-

arXiv:2109.03744v1
fatcat:gb2yhebbpve4ppr2kflqylb2fm
*regular*,*bipartite**graphs*, runs in time exp( O( n ·log^3 d /d ) ),*and*outputs a (1 + o(1))-approximation to the number of*independent**sets*. ... Our first algorithm applies to d-*regular*,*bipartite**graphs*satisfying a weak expansion condition: when d is constant,*and*the*graph*is a*bipartite*Ω( log^2 d/d)-expander, we obtain an FPTAS for the number ... Previous results*on*expander*graphs*include an FPTAS for i(G) in the case that G is a typical d-*regular*,*random**bipartite**graph*[31, 40, 8] . ...##
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Page 7571 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

Summary: “Let A

*and*B be two*sets*of nm points in the plane,*and*let M be a (*one*-to-*one*) matching between A*and*B. ... matchings*on*2n vertices*and*conditioning*on*the result being an r-*regular**graph*(i.e., conditioning*on*the event that all matchings are dis- joint). ...
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