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Greedy maximal independent sets via local limits Peleg Michaeli Tel Aviv University Workshop on Local Algorithms WOLA 2019 ETH Zurich, July 21, 2019 Joint work with Michael Krivelevich, Tams Mszros and Clara Shikhelman Peleg Michaeli

  1. General framework Decay of correlation = a.a.s. Develop a machinery to calculate the probability that the root is red. Peleg Michaeli (TAU) Greedy MIS July 21, 2019 7 / 18 We wish to calculate the asymptotics of ι ( G n ) . We fjrst calculate E ( ι ( G n )) = P ( ρ n ∈ I ( G n )) for ρ n chosen u.a.r. We hope that this is determined by a small neighbourhood of ρ n . This local view of ρ n is captured by the local limit of G n .

  2. General framework Decay of correlation = Develop a machinery to calculate the probability that the root is red. Peleg Michaeli (TAU) Greedy MIS July 21, 2019 7 / 18 We wish to calculate the asymptotics of ι ( G n ) . We fjrst calculate E ( ι ( G n )) = P ( ρ n ∈ I ( G n )) for ρ n chosen u.a.r. We hope that this is determined by a small neighbourhood of ρ n . This local view of ρ n is captured by the local limit of G n . ⇒ ι ( G n ) ∼ E ( ι ( G n )) a.a.s.

  3. General framework Decay of correlation = Develop a machinery to calculate the probability that the root is red. Peleg Michaeli (TAU) Greedy MIS July 21, 2019 7 / 18 We wish to calculate the asymptotics of ι ( G n ) . We fjrst calculate E ( ι ( G n )) = P ( ρ n ∈ I ( G n )) for ρ n chosen u.a.r. We hope that this is determined by a small neighbourhood of ρ n . This local view of ρ n is captured by the local limit of G n . ⇒ ι ( G n ) ∼ E ( ι ( G n )) a.a.s.

  4. Local limits Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18 We say that a (random) graph sequence G n locally converges to a random rooted graph ( U, ρ ) , if for every r ≥ 0 , the ball B r ( G, ρ n ) converges in distribution to B r ( U, ρ ) , where ρ n is a uniform vertex of G n .

  5. Local limits Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18 We say that a (random) graph sequence G n locally converges to a random rooted graph ( U, ρ ) , if for every r ≥ 0 , the ball B r ( G, ρ n ) converges in distribution to B r ( U, ρ ) , where ρ n is a uniform vertex of G n .

  6. Local limits Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18 We say that a (random) graph sequence G n locally converges to a random rooted graph ( U, ρ ) , if for every r ≥ 0 , the ball B r ( G, ρ n ) converges in distribution to B r ( U, ρ ) , where ρ n is a uniform vertex of G n .

  7. Local limits Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18 We say that a (random) graph sequence G n locally converges to a random rooted graph ( U, ρ ) , if for every r ≥ 0 , the ball B r ( G, ρ n ) converges in distribution to B r ( U, ρ ) , where ρ n is a uniform vertex of G n .

  8. Local limits Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18 We say that a (random) graph sequence G n locally converges to a random rooted graph ( U, ρ ) , if for every r ≥ 0 , the ball B r ( G, ρ n ) converges in distribution to B r ( U, ρ ) , where ρ n is a uniform vertex of G n .

  9. Local limits Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18 We say that a (random) graph sequence G n locally converges to a random rooted graph ( U, ρ ) , if for every r ≥ 0 , the ball B r ( G, ρ n ) converges in distribution to B r ( U, ρ ) , where ρ n is a uniform vertex of G n . · · ·

  10. Peleg Michaeli (TAU) Local limits Greedy MIS July 21, 2019 8 / 18 We say that a (random) graph sequence G n locally converges to a random rooted graph ( U, ρ ) , if for every r ≥ 0 , the ball B r ( G, ρ n ) converges in distribution to B r ( U, ρ ) , where ρ n is a uniform vertex of G n . · · · � ➪ ➪

  11. Peleg Michaeli (TAU) Local limits Greedy MIS July 21, 2019 8 / 18 We say that a (random) graph sequence G n locally converges to a random rooted graph ( U, ρ ) , if for every r ≥ 0 , the ball B r ( G, ρ n ) converges in distribution to B r ( U, ρ ) , where ρ n is a uniform vertex of G n . · · · � ➪ ➪ � ➪ ➪

  12. Local limits loc July 21, 2019 Greedy MIS Peleg Michaeli (TAU) loc 8 / 18 Examples loc We say that a (random) graph sequence G n locally converges to a random rooted graph ( U, ρ ) , if for every r ≥ 0 , the ball B r ( G, ρ n ) converges in distribution to B r ( U, ρ ) , where ρ n is a uniform vertex of G n . P n , C n − → Z [ n ] d loc → Z d − G ( n, λ / n ) loc − → T λ , a Galton-Watson Pois ( λ ) 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 Theorem (Krivelevich, Mészáros, M., Shikhelman ’19+) loc Peleg Michaeli (TAU) Greedy MIS July 21, 2019 9 / 18 Suppose G n has subfactorial growth. − → ( U, ρ ) then ι ( G n ) → ι ( U, ρ ) a.a.s. If G n

  14. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  15. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  16. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  17. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  18. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  19. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  20. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  21. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  22. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  23. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  24. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  25. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  26. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  27. Decay of correlation Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

  28. Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., Locally tree-like but even is still unknown... graph sequences for which is a.s. a tree. Children of the past are roots to independent subtrees. Peleg Michaeli (TAU) Greedy MIS July 21, 2019 11 / 18 We need to calculate ι ( U, ρ ) ,

  29. Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., Locally tree-like graph sequences for which is a.s. a tree. Children of the past are roots to independent subtrees. Peleg Michaeli (TAU) Greedy MIS July 21, 2019 11 / 18 We need to calculate ι ( U, ρ ) , but even ι ( Z 2 ) is still unknown...

  30. Locally tree-like Children of the past are roots to independent subtrees. Peleg Michaeli (TAU) Greedy MIS July 21, 2019 11 / 18 We need to calculate ι ( U, ρ ) , but even ι ( Z 2 ) is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., graph sequences for which ( U, ρ ) is a.s. a tree.

  31. Locally tree-like Children of the past are roots to independent subtrees. Peleg Michaeli (TAU) Greedy MIS July 21, 2019 11 / 18 We need to calculate ι ( U, ρ ) , but even ι ( Z 2 ) is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., graph sequences for which ( U, ρ ) is a.s. a tree. u 1 u 2 · · · u d ρ

  32. Locally tree-like Children of the past are roots to independent subtrees. Peleg Michaeli (TAU) Greedy MIS July 21, 2019 11 / 18 We need to calculate ι ( U, ρ ) , but even ι ( Z 2 ) is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., graph sequences for which ( U, ρ ) is a.s. a tree. u 1 u 2 · · · u d ρ

  33. Systems of ordinary difgerential equations I I I I Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18 Let ( U, ρ ) be a single-type branching process.

  34. Systems of ordinary difgerential equations I I I Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18 Let ( U, ρ ) be a single-type branching process. y ( x ) = P ( ρ ∈ I ( U, ρ ) ∧ σ ρ < x )

  35. I Systems of ordinary difgerential equations Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18 Let ( U, ρ ) be a single-type branching process. y ( x ) = P ( ρ ∈ I ( U, ρ ) ∧ σ ρ < x ) = x · P ( ρ ∈ I ( U, ρ ) | σ ρ < x ) ∫ x = P ( ρ ∈ I ( U, ρ ) | σ ρ = z ) dz 0

  36. Systems of ordinary difgerential equations Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18 Let ( U, ρ ) be a single-type branching process. y ( x ) = P ( ρ ∈ I ( U, ρ ) ∧ σ ρ < x ) = x · P ( ρ ∈ I ( U, ρ ) | σ ρ < x ) ∫ x = P ( ρ ∈ I ( U, ρ ) | σ ρ = z ) dz 0 y ′ ( x ) = P ( ρ ∈ I ( U, ρ ) | σ ρ = x )

  37. Systems of ordinary difgerential equations Peleg Michaeli (TAU) July 21, 2019 Greedy MIS 12 / 18 Let ( U, ρ ) be a single-type branching process. y ( x ) = P ( ρ ∈ I ( U, ρ ) ∧ σ ρ < x ) = x · P ( ρ ∈ I ( U, ρ ) | σ ρ < x ) ∫ x = P ( ρ ∈ I ( U, ρ ) | σ ρ = z ) dz 0 y ′ ( x ) = P ( ρ ∈ I ( U, ρ ) | σ ρ = x ) Thus, if y is a unique solution of ) ℓ )( 1 − y ( x ) ξ <x = ℓ ∑ y ′ ( x ) = ( y (0) = 0 , P , x ℓ ∈ N then, ι ( U, ρ ) = y (1) .

  38. Systems of ordinary difgerential equations Peleg Michaeli (TAU) July 21, 2019 Greedy MIS 12 / 18 Let ( U, ρ ) be a multi-type branching process. y ( x ) = P ( ρ ∈ I ( U, ρ ) ∧ σ ρ < x ) = x · P ( ρ ∈ I ( U, ρ ) | σ ρ < x ) ∫ x = P ( ρ ∈ I ( U, ρ ) | σ ρ = z ) dz 0 y ′ ( x ) = P ( ρ ∈ I ( U, ρ ) | σ ρ = x ) Thus, if y is a unique solution of ) ℓ j )( 1 − y j ( x ) ( ∑ ∏ ξ <x y ′ k ( x ) = k → j = ℓ j y k (0) = 0 , P , x ℓ ∈ N T j ∈ T then, ι ( U, ρ ) = E ( y k (1)) .

  39. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  40. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  41. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  42. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  43. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  44. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  45. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  46. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  47. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  48. Size-biased Galton-Watson branching processes Kolchin, Grimmett: the sequence of uniform random trees locally Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18 converges to the size-biased Galton-Watson Pois (1) tree.

  49. Uniform random trees , s t s hence s , and for s s , and we get s Peleg Michaeli (TAU) Greedy MIS July 21, 2019 t s 14 / 18 ∞ ) d ( λx ) d ( 1 − y t ( x ) ∑ = e − λy t ( x ) . y ′ t ( x ) = e λx d ! x d =0 hence y t ( x ) = ln (1 + λx )/ λ . Thus ι ( G ( n, λ / n )) → ι ( T λ ) = y t (1) = ln (1 + λ ) . λ

  50. Uniform random trees and we get July 21, 2019 Greedy MIS Peleg Michaeli (TAU) 14 / 18 ∞ ) d ( λx ) d ( 1 − y t ( x ) ∑ = e − λy t ( x ) . y ′ t ( x ) = e λx d ! x d =0 hence y t ( x ) = ln (1 + λx )/ λ . Thus ι ( G ( n, λ / n )) → ι ( T λ ) = y t (1) = ln (1 + λ ) . λ t ( x ) = (1 − y s ( x )) e − λy t ( x ) = 1 − y s ( x ) y ′ s ( x ) = (1 − y s ( x )) y ′ 1 + λx , hence y s ( x ) = 1 − (1 + λx ) − 1/ λ , and for λ = 1 , y s (1) = 1 − (1 + x ) − 1 , T 1 ) = y s (1) = 1 ι ( T n ) → ι ( ˆ 2 .

  51. Simulations don’t lie red: 125 (50%), green: 92 ( 37%), blue: 32 ( 13%), black: 1 Peleg Michaeli (TAU) Greedy MIS July 21, 2019 15 / 18

  52. Simulations don’t lie Peleg Michaeli (TAU) Greedy MIS July 21, 2019 15 / 18 red: 125 (50%), green: 92 ( ≈ 37%), blue: 32 ( ≈ 13%), black: 1

  53. Simulations don’t lie (but I do) Peleg Michaeli (TAU) Greedy MIS July 21, 2019 15 / 18 red: 125 (50%), green: 92 ( ≈ 37%), blue: 32 ( ≈ 13%), black: 1

  54. Greedy independence ratio – results Flory ’39, Page ’59 July 21, 2019 Greedy MIS Peleg Michaeli (TAU) (same for functional digraphs) KMMS ’19+ Lauer & Wormald ’07 16 / 18 McDiarmid ’84 Wormald ’95 ι ( P n ) → 1 2 (1 − e − 2 ) ι ( G ( n, λ / n )) → ln (1 + λ )/ λ ι ( G n,d ) → 1 ( 1 − ( d − 1) − 2/( d − 2) ) 2 ( d -regular graphs with girth → ∞ )

  55. Greedy independence ratio – results Flory ’39, Page ’59 July 21, 2019 Greedy MIS Peleg Michaeli (TAU) (same for functional digraphs) KMMS ’19+ Lauer & Wormald ’07 16 / 18 McDiarmid ’84 Wormald ’95 ✓ ι ( P n ) → 1 2 (1 − e − 2 ) ι ( G ( n, λ / n )) → ln (1 + λ )/ λ ι ( G n,d ) → 1 ( 1 − ( d − 1) − 2/( d − 2) ) 2 ( d -regular graphs with girth → ∞ )

  56. Greedy independence ratio – results Flory ’39, Page ’59 July 21, 2019 Greedy MIS Peleg Michaeli (TAU) (same for functional digraphs) KMMS ’19+ Lauer & Wormald ’07 16 / 18 Wormald ’95 McDiarmid ’84 ✓ ι ( P n ) → 1 2 (1 − e − 2 ) ✓ ι ( G ( n, λ / n )) → ln (1 + λ )/ λ ι ( G n,d ) → 1 ( 1 − ( d − 1) − 2/( d − 2) ) 2 ( d -regular graphs with girth → ∞ )

  57. Greedy independence ratio – results Flory ’39, Page ’59 July 21, 2019 Greedy MIS Peleg Michaeli (TAU) (same for functional digraphs) KMMS ’19+ Lauer & Wormald ’07 16 / 18 Wormald ’95 McDiarmid ’84 ✓ ι ( P n ) → 1 2 (1 − e − 2 ) ✓ ι ( G ( n, λ / n )) → ln (1 + λ )/ λ 1 − ( d − 1) − 2/( d − 2) ) ✓ ι ( G n,d ) → 1 ( 2 ( d -regular graphs with girth → ∞ )

  58. Greedy independence ratio – results Flory ’39, Page ’59 July 21, 2019 Greedy MIS Peleg Michaeli (TAU) (same for functional digraphs) KMMS ’19+ Lauer & Wormald ’07 16 / 18 Wormald ’95 McDiarmid ’84 ✓ ι ( P n ) → 1 2 (1 − e − 2 ) ✓ ι ( G ( n, λ / n )) → ln (1 + λ )/ λ 1 − ( d − 1) − 2/( d − 2) ) ✓ ι ( G n,d ) → 1 ( 2 ( d -regular graphs with girth → ∞ ) ✓

  59. Greedy independence ratio – results Flory ’39, Page ’59 July 21, 2019 Greedy MIS Peleg Michaeli (TAU) (same for functional digraphs) KMMS ’19+ Lauer & Wormald ’07 16 / 18 McDiarmid ’84 Wormald ’95 ✓ ι ( P n ) → 1 2 (1 − e − 2 ) ✓ ι ( G ( n, λ / n )) → ln (1 + λ )/ λ 1 − ( d − 1) − 2/( d − 2) ) ✓ ι ( G n,d ) → 1 ( 2 ( d -regular graphs with girth → ∞ ) ✓ ✸ ι ( T n ) → 1 2

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