Approximately counting independent sets in bipartite graphs via graph containers

By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to d$$ d $$‐regular, bipartite graphs satisfying a weak expansion condition: when d$$ d $$ is co...

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Bibliographic Details
Published in:Random structures & algorithms Vol. 63; no. 1; pp. 215 - 241
Main Authors: Jenssen, Matthew, Perkins, Will, Potukuchi, Aditya
Format: Journal Article
Language:English
Published: New York John Wiley & Sons, Inc 01.08.2023
Wiley Subscription Services, Inc
Subjects:
Algorithms
approximate counting
Approximation
Codes
Containers
Fugacity
graph containers
Graph theory
Graphs
independent sets
Partitions (mathematics)
ISSN:1042-9832, 1098-2418
Online Access:Get full text
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Summary:By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to d$$ d $$‐regular, bipartite graphs satisfying a weak expansion condition: when d$$ d $$ is constant, and the graph is a bipartite Ω(log2d/d)$$ \Omega \left({\log}^2d/d\right) $$‐expander, we obtain an FPTAS for the number of independent sets. Previously such a result for d>5$$ d>5 $$ was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: for a d$$ d $$‐regular, bipartite α$$ \alpha $$‐expander, with α>0$$ \alpha >0 $$ fixed, we give an FPTAS for the hard‐core model partition function at fugacity λ=Ω(logd/d1/4)$$ \lambda =\Omega \left(\log d/{d}^{1/4}\right) $$. Finally we present an algorithm that applies to all d$$ d $$‐regular, bipartite graphs, runs in time expOn·log3dd$$ \exp \left(O\left(n\cdotp \frac{\log^3d}{d}\right)\right) $$, and outputs a (1+o(1))$$ \left(1+o(1)\right) $$‐approximation to the number of independent sets.
Bibliography:Funding information
NSF,Grant/Award Numbers: DMS‐1847451;CCF‐1934915
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.21145