greedy maximal independent sets via local limits
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Greedy maximal independent sets via local limits Peleg Michaeli Tel - PowerPoint PPT Presentation

Greedy maximal independent sets via local limits Peleg Michaeli Tel Aviv University The 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA2020) September 2020 Joint work


  1. is captured by the local limit of local limit is red. General framework Decay of correlation = This local view of . Develop a machinery to calculate the probability that the root of the Let G n be a graph sequence satisfying | G n | → ∞ . We wish to calculate the asymptotics of ι ( G n ) . We approximate E [ ι ( G n )] = P ( ρ n ∈ I ( G n )) for ρ n chosen uniformly. We hope that this is determined by a small neighbourhood of ρ n . ⇒ ι ( G n ) ∼ E [ ι ( G n )] a.a.s.

  2. local limit is red. General framework Decay of correlation = Develop a machinery to calculate the probability that the root of the Let G n be a graph sequence satisfying | G n | → ∞ . We wish to calculate the asymptotics of ι ( G n ) . We approximate E [ ι ( G n )] = P ( ρ n ∈ I ( G n )) for ρ n chosen uniformly. We hope that this is determined by a small neighbourhood of ρ n . ⇒ ι ( G n ) ∼ E [ ι ( G n )] a.a.s. This local view of ρ n is captured by the local limit of G n .

  3. General framework Decay of correlation = Develop a machinery to calculate the probability that the root of the local limit is red. Let G n be a graph sequence satisfying | G n | → ∞ . We wish to calculate the asymptotics of ι ( G n ) . We approximate E [ ι ( G n )] = P ( ρ n ∈ I ( G n )) for ρ n chosen uniformly. We hope that this is determined by a small neighbourhood of ρ n . ⇒ ι ( G n ) ∼ E [ ι ( G n )] a.a.s. This local view of ρ n is captured by the local limit of G n .

  4. Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .

  5. Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .

  6. Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .

  7. Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .

  8. Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .

  9. Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n . · · ·

  10. Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n . · · · � ➪ ➪

  11. Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n . · · · � ➪ ➪ � ➪ ➪

  12. Local limits (a.k.a. Benjamini–Schramm Limits) loc Examples loc loc We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n . P n , C n − → Z [ n ] d loc → Z d − G ( n, d / n ) loc − → T d , a Galton–Watson Pois ( d ) tree − → the d -regular tree G n,d → ˆ Uniform random tree T n − T 1 , a size-biased GW Pois (1) tree − → the canopy tree Finite d -ary balanced tree loc

  13. Convergence of the greedy independence ratio from a typical vertex is subfactorial in their length. (bounded degree subfactorial path growth) Theorem (Krivelevich, Mészáros, M., Shikhelman ’20) Suppose has subfactorial path growth. If loc then a.a.s. is red Say that G n has subfactorial path growth if the expected number of paths

  14. Convergence of the greedy independence ratio from a typical vertex is subfactorial in their length. Theorem (Krivelevich, Mészáros, M., Shikhelman ’20) Suppose has subfactorial path growth. If loc then a.a.s. is red Say that G n has subfactorial path growth if the expected number of paths (bounded degree ⊊ subfactorial path growth)

  15. Convergence of the greedy independence ratio from a typical vertex is subfactorial in their length. Theorem (Krivelevich, Mészáros, M., Shikhelman ’20) loc is red Say that G n has subfactorial path growth if the expected number of paths (bounded degree ⊊ subfactorial path growth) Suppose G n has subfactorial path growth. If G n − → ( U, ρ ) then ι ( G n ) → ι ( U, ρ ) a.a.s.

  16. Convergence of the greedy independence ratio from a typical vertex is subfactorial in their length. Theorem (Krivelevich, Mészáros, M., Shikhelman ’20) loc Say that G n has subfactorial path growth if the expected number of paths (bounded degree ⊊ subfactorial path growth) Suppose G n has subfactorial path growth. If G n − → ( U, ρ ) then ι ( G n ) → ι ( U, ρ ) a.a.s. P ( ρ is red )

  17. Exploration algorithms / decay of correlation

  18. Exploration algorithms / decay of correlation

  19. Exploration algorithms / decay of correlation

  20. Exploration algorithms / decay of correlation

  21. Exploration algorithms / decay of correlation

  22. Exploration algorithms / decay of correlation

  23. Exploration algorithms / decay of correlation

  24. Exploration algorithms / decay of correlation

  25. Exploration algorithms / decay of correlation

  26. Exploration algorithms / decay of correlation

  27. Exploration algorithms / decay of correlation

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