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

Chao Liao, Jiabao Lin, Pinyan Lu, Zhenyu Mao
2019 arXiv   pre-print
We give a fully polynomial-time approximation scheme (FPTAS) to 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] .  ... 
arXiv:1903.07531v1 fatcat:enbkxxpa2vdz7p754dwz235zjq

Counting Independent Sets and Colorings on Random Regular Bipartite Graphs

Chao Liao, Jiabao Lin, Pinyan Lu, Zhenyu Mao, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We give a fully polynomial-time approximation scheme (FPTAS) to 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.  ... 
doi:10.4230/lipics.approx-random.2019.34 dblp:conf/approx/LiaoLLM19 fatcat:ip6tpov4sjbb3a5k54pxpvzlbm

Algorithms for #BIS-hard problems on expander graphs [chapter]

Matthew Jenssen, Peter Keevash, Will Perkins
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 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.  ... 
doi:10.1137/1.9781611975482.135 dblp:conf/soda/JenssenKP19 fatcat:nvvcwyd7d5hj3ashc27iphf4aa

Algorithms for #BIS-Hard Problems on Expander Graphs

Matthew Jenssen, Peter Keevash, Will Perkins
2020 SIAM journal on computing (Print)  
This would furnish us with an FPTAS for 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  ... 
doi:10.1137/19m1286669 fatcat:cvn3rdusmncibiansjubwpz434

Counting Colorings of a Regular Graph

David Galvin
2014 Graphs and Combinatorics  
• The 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.  ... 
doi:10.1007/s00373-013-1403-z fatcat:qxets5cmk5cp5ijzvn3u66qjme

Sidorenko's conjecture, colorings and independent sets [article]

Péter Csikvári, Zhicong Lin
2017 arXiv   pre-print
These cases correspond to 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.  ... 
arXiv:1603.05888v4 fatcat:a5xccocz2jhczdd2e4mgtnlw54

Sidorenko's Conjecture, Colorings and Independent Sets

Péter Csikvári, Zhicong Lin
2017 Electronic Journal of Combinatorics  
These cases correspond to 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  ... 
doi:10.37236/6019 fatcat:vlfw7vy6kzg6vkbaztagwidgm4

Algorithms for #BIS-hard problems on expander graphs [article]

Matthew Jenssen, Peter Keevash, Will Perkins
2020 arXiv   pre-print
We also find efficient 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.  ... 
arXiv:1807.04804v3 fatcat:bhldiwazznazpokcqnclo6xrde

Three tutorial lectures on entropy and counting [article]

David Galvin
2014 arXiv   pre-print
We explain the notion of the entropy of a discrete 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;  ... 
arXiv:1406.7872v1 fatcat:cvqnryzj5jfnphyooktpqmhcae

On Markov Chains for Randomly H-Coloring a Graph

Colin Cooper, Martin Dyer, Alan Frieze
2001 Journal of Algorithms  
The probabilistic model we consider includes 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.  ... 
doi:10.1006/jagm.2000.1142 fatcat:6wg4j2ekvngqtjqrlkpgsfc6fy

1-factorizations of pseudorandom graphs [article]

Asaf Ferber, Vishesh Jain
2018 arXiv   pre-print
Specifically, we prove that an (n,d,λ)-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)}.  ... 
arXiv:1803.10361v1 fatcat:src4qm5kazd6pixztnwh2bzczm

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  ... 

Counting dominating sets and related structures in graphs [article]

Jonathan Cutler, A. J. Radcliffe
2015 arXiv   pre-print
We consider some problems concerning the maximum number of (strong) dominating 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) .  ... 
arXiv:1503.00998v1 fatcat:zrxingtkwfadlnqgnu5miffgxe

Approximately counting independent sets in bipartite graphs via graph containers [article]

Matthew Jenssen and Will Perkins and Aditya Potukuchi
2021 arXiv   pre-print
Finally we present an algorithm that applies to all d-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] .  ... 
arXiv:2109.03744v1 fatcat:gb2yhebbpve4ppr2kflqylb2fm

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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