Polymer Dynamics via Cliques: New Conditions for Approximations

Friedrich, Tobias; Göbel, Andreas; Krejca, Martin S.; Pappik, Marcus

doi:10.48550/arxiv.2007.08293

Cited by 4 publications

(14 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [FGKP20], the authors use a different Markov chain condition, the 'clique dynamics condition', very similar to (6), which also requires requires weights of polymers of size k to decay like (eq∆) −k , saving the same factor e over (5). Their running times, though, are again of the form n O(log ∆) since implementing one step of their Markov chain involves enumerating rooted polymers of size O(log n).…”

Section: Sampling From Polymer Modelsmentioning

confidence: 99%

“…then there is a perfect sampling algorithm for µ hc G running in expected time O(n log n). Approximate sampling algorithms with very large polynomial run times were previously given for this problem when 6λ∆ L ∆ R < (1 + λ) δ R /∆ L in [CP20], and later in [FGKP20] when 3.3353λ∆ L ∆ R < (1 + λ) δ R /∆ L . Our result applies for a comparable parameter range, and is the first to achieve perfect sampling and near-linear running time.…”

Section: Hard-core Model On Bipartite Graphsmentioning

confidence: 99%

See 1 more Smart Citation

Fast and perfect sampling of subgraphs and polymer systems

Blanca¹,

Cannon²,

Perkins³

2022

Preprint

View full text Add to dashboard Cite

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs under a vertex-percolation subcriticality condition. We show that this subcriticality condition is optimal in the sense that the problem of (approximately) sampling weighted rooted graphlets becomes impossible for infinite graphs and intractable for finite graphs if the condition does not hold. We apply our rooted graphlet sampling algorithm as a subroutine to give a fast perfect sampling algorithm for polymer models and a fast perfect sampling algorithm for weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. We apply this polymer model algorithm to give improved sampling algorithms for spin systems at low temperatures on expander graphs and other structured families of graphs: under the least restrictive conditions known we give near linear-time algorithms, while previous algorithms in these regimes required large polynomial running times.

show abstract

Section: Sampling From Polymer Modelsmentioning

confidence: 99%

Section: Hard-core Model On Bipartite Graphsmentioning

confidence: 99%

Fast and perfect sampling of subgraphs and polymer systems

Blanca¹,

Cannon²,

Perkins³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Then the cluster expansion converges absolutely. Moreover, for every vertex v, (12) Γ∈C(G) Γ∋v φ(Γ) γ∈Γ w γ e g(Γ) ≤ 1.…”

Section: Polymer Modelsmentioning

confidence: 99%

“…Another line of work in this direction includes [31] in which approximation algorithms are given for the hard-core partition function Z G (λ) (counting weighted independent sets) in bounded-degree, bipartite expander graphs, based on two tools from statistical physics: polymer models and the cluster expansion (following [28]). This work was followed by several improvements, extensions, and generalizations including [40,5,7,13,12,14,8]. All of these algorithms are 'low-temperature' algorithms: they exploit the fact that on a bipartite graph with sufficient expansion, most (weighted) independent sets have few vertices from one side of the bipartition; that is, they are close to one of the two ground states consisting of all subsets of one side.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Approximately counting independent sets in bipartite graphs via graph containers

Jenssen¹,

Perkins²,

Potukuchi³

2021

Preprint

View full text Add to dashboard Cite

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-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 of independent sets. Previously such a result for 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-regular, bipartite α-expander, with α > 0 fixed, we give an FPTAS for the hard-core model partition function at fugacity λ = Ω(log d/d 1/4 ). 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.

show abstract

Approximately counting independent sets in bipartite graphs via graph containers

Jenssen

Perkins

Potukuchi

2023

Random Struct Algorithms

View full text Add to dashboard Cite

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-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 of independent sets. Previously such a result for 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-regular, bipartite 𝛼-expander, with 𝛼 > 0 fixed, we give an FPTAS for the hard-core model partition function at fugacity 𝜆 = Ω(log 𝑑∕𝑑 1∕4 ). Finally we present an algorithm that applies to all 𝑑-regular, bipartite graphs, runs in time exp, and outputs a (1+o(1))-approximation to the number of independent sets.

show abstract

Polymer Dynamics via Cliques: New Conditions for Approximations

Cited by 4 publications

References 0 publications

Fast and perfect sampling of subgraphs and polymer systems

Fast and perfect sampling of subgraphs and polymer systems

Approximately counting independent sets in bipartite graphs via graph containers

Approximately counting independent sets in bipartite graphs via graph containers

Contact Info

Product

Resources

About