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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Vydáno v:Random structures & algorithms Ročník 63; číslo 1; s. 215 - 241
Hlavní autoři: Jenssen, Matthew, Perkins, Will, Potukuchi, Aditya
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York John Wiley & Sons, Inc 01.08.2023
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Témata:
Algorithms
approximate counting
Approximation
Codes
Containers
Fugacity
graph containers
Graph theory
Graphs
independent sets
Partitions (mathematics)
ISSN:1042-9832, 1098-2418
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Shrnutí: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.
Bibliografie:Funding information
NSF,Grant/Award Numbers: DMS‐1847451;CCF‐1934915
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.21145