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

Chao Liao, Jiabao Lin, Pinyan Lu, Zhenyu Mao
<span title="2019-03-18">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1903.07531v1">arXiv:1903.07531v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/enbkxxpa2vdz7p754dwz235zjq">fatcat:enbkxxpa2vdz7p754dwz235zjq</a> </span>
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Counting Independent Sets and Colorings on Random Regular Bipartite Graphs

Chao Liao, Jiabao Lin, Pinyan Lu, Zhenyu Mao, Michael Wagner
<span title="2019-09-19">2019</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gdc7gjvgv5hd7mr5o3uwxvngwu" style="color: black;">International Workshop on Approximation Algorithms for Combinatorial Optimization</a> </i> &nbsp;
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.approx-random.2019.34">doi:10.4230/lipics.approx-random.2019.34</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/approx/LiaoLLM19.html">dblp:conf/approx/LiaoLLM19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ip6tpov4sjbb3a5k54pxpvzlbm">fatcat:ip6tpov4sjbb3a5k54pxpvzlbm</a> </span>
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Algorithms for #BIS-hard problems on expander graphs [chapter]

Matthew Jenssen, Peter Keevash, Will Perkins
<span title="">2019</span> <i title="Society for Industrial and Applied Mathematics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ja7dpjgz2nhjlovlil3ij5m7ei" style="color: black;">Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms</a> </i> &nbsp;
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/1.9781611975482.135">doi:10.1137/1.9781611975482.135</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/soda/JenssenKP19.html">dblp:conf/soda/JenssenKP19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nvvcwyd7d5hj3ashc27iphf4aa">fatcat:nvvcwyd7d5hj3ashc27iphf4aa</a> </span>
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Algorithms for #BIS-Hard Problems on Expander Graphs

Matthew Jenssen, Peter Keevash, Will Perkins
<span title="">2020</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7dys7zoberdktmxyjciuy5bnse" style="color: black;">SIAM journal on computing (Print)</a> </i> &nbsp;
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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/19m1286669">doi:10.1137/19m1286669</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cvn3rdusmncibiansjubwpz434">fatcat:cvn3rdusmncibiansjubwpz434</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20201106150639/http://pure-oai.bham.ac.uk/ws/files/90892233/1.9781611975482.135.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5f/df/5fdffc760318664a880a8ff1b2946156006c0db2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/19m1286669"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Counting Colorings of a Regular Graph

David Galvin
<span title="2014-01-14">2014</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yxooi3wjmbgqtbp7l4evjeuj4i" style="color: black;">Graphs and Combinatorics</a> </i> &nbsp;
• 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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00373-013-1403-z">doi:10.1007/s00373-013-1403-z</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qxets5cmk5cp5ijzvn3u66qjme">fatcat:qxets5cmk5cp5ijzvn3u66qjme</a> </span>
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Sidorenko's conjecture, colorings and independent sets [article]

Péter Csikvári, Zhicong Lin
<span title="2017-02-02">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1603.05888v4">arXiv:1603.05888v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/a5xccocz2jhczdd2e4mgtnlw54">fatcat:a5xccocz2jhczdd2e4mgtnlw54</a> </span>
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Sidorenko's Conjecture, Colorings and Independent Sets

Péter Csikvári, Zhicong Lin
<span title="2017-01-20">2017</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/6019">doi:10.37236/6019</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vlfw7vy6kzg6vkbaztagwidgm4">fatcat:vlfw7vy6kzg6vkbaztagwidgm4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200322234724/http://dspace.mit.edu/bitstream/handle/1721.1/110146/Sidorenko&#39;s%20conjecture.pdf;jsessionid=49A4B7D41D2031FAF539A67AFE5C17FB?sequence=1" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ee/32/ee323d727c5022c08d427ff00f6c53af356d9faf.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/6019"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

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

Matthew Jenssen, Peter Keevash, Will Perkins
<span title="2020-03-23">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1807.04804v3">arXiv:1807.04804v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bhldiwazznazpokcqnclo6xrde">fatcat:bhldiwazznazpokcqnclo6xrde</a> </span>
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Three tutorial lectures on entropy and counting [article]

David Galvin
<span title="2014-06-30">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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;  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1406.7872v1">arXiv:1406.7872v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cvqnryzj5jfnphyooktpqmhcae">fatcat:cvqnryzj5jfnphyooktpqmhcae</a> </span>
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On Markov Chains for Randomly H-Coloring a Graph

Colin Cooper, Martin Dyer, Alan Frieze
<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/anp7kqfyerdljfxydqb4objj6a" style="color: black;">Journal of Algorithms</a> </i> &nbsp;
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jagm.2000.1142">doi:10.1006/jagm.2000.1142</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6wg4j2ekvngqtjqrlkpgsfc6fy">fatcat:6wg4j2ekvngqtjqrlkpgsfc6fy</a> </span>
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1-factorizations of pseudorandom graphs [article]

Asaf Ferber, Vishesh Jain
<span title="2018-03-28">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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)}.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1803.10361v1">arXiv:1803.10361v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/src4qm5kazd6pixztnwh2bzczm">fatcat:src4qm5kazd6pixztnwh2bzczm</a> </span>
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Page 4037 of Mathematical Reviews Vol. , Issue 2003f [page]

<span title="">2003</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
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]

Jonathan Cutler, A. J. Radcliffe
<span title="2015-03-03">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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) .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1503.00998v1">arXiv:1503.00998v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zrxingtkwfadlnqgnu5miffgxe">fatcat:zrxingtkwfadlnqgnu5miffgxe</a> </span>
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Approximately counting independent sets in bipartite graphs via graph containers [article]

Matthew Jenssen and Will Perkins and Aditya Potukuchi
<span title="2021-09-08">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2109.03744v1">arXiv:2109.03744v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gb2yhebbpve4ppr2kflqylb2fm">fatcat:gb2yhebbpve4ppr2kflqylb2fm</a> </span>
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Page 7571 of Mathematical Reviews Vol. , Issue 98M [page]

<span title="">1998</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
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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