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

▶
greedy maximal independent sets via local limits
SMART_READER_LITE
LIVE PREVIEW

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

Nov 01, 2023 481 likes •1.57k views

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

slide-1
SLIDE 1

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, Tamás Mészáros and Clara Shikhelman

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 1 / 18

slide-2
SLIDE 2

Independent sets

Finding maximum independent sets is very hard Finding maximal independent sets is very easy

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 2 / 18

slide-3
SLIDE 3

Independent sets

Finding maximum independent sets is very hard Finding maximal independent sets is very easy

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 2 / 18

slide-4
SLIDE 4

Independent sets

Finding maximum independent sets is very hard Finding maximal independent sets is very easy

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 2 / 18

slide-5
SLIDE 5

Random greedy MIS – sequential

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-6
SLIDE 6

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-7
SLIDE 7

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-8
SLIDE 8

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-9
SLIDE 9

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-10
SLIDE 10

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-11
SLIDE 11

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-12
SLIDE 12

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-13
SLIDE 13

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-14
SLIDE 14

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-15
SLIDE 15

Random greedy MIS – sequential

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 3 / 18

slide-16
SLIDE 16

Random greedy MIS – parallel

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 4 / 18

slide-17
SLIDE 17

Random greedy MIS – parallel

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 4 / 18

slide-18
SLIDE 18

Random greedy MIS – parallel

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 4 / 18

slide-19
SLIDE 19

Random greedy MIS – parallel

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 4 / 18

slide-20
SLIDE 20

Random greedy MIS – parallel

1 2 3 4 5 6 7 8 9

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 4 / 18

slide-21
SLIDE 21

Greedy independence ratio – previous work

Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. Flory ’39, Page ’59 McDiarmid ’84 ln Wormald ’95 Lauer & Wormald ’07 (same for

  • regular graphs with girth

) BJL ’17, BJP ’17

  • f random graphs with given degree sequence

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 5 / 18

slide-22
SLIDE 22

Greedy independence ratio – previous work

Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

McDiarmid ’84 ln Wormald ’95 Lauer & Wormald ’07 (same for

  • regular graphs with girth

) BJL ’17, BJP ’17

  • f random graphs with given degree sequence

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 5 / 18

slide-23
SLIDE 23

Greedy independence ratio – previous work

Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ Wormald ’95 Lauer & Wormald ’07 (same for

  • regular graphs with girth

) BJL ’17, BJP ’17

  • f random graphs with given degree sequence

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 5 / 18

slide-24
SLIDE 24

Greedy independence ratio – previous work

Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) Lauer & Wormald ’07 (same for

  • regular graphs with girth

) BJL ’17, BJP ’17

  • f random graphs with given degree sequence

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 5 / 18

slide-25
SLIDE 25

Greedy independence ratio – previous work

Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) Lauer & Wormald ’07 (same for d-regular graphs with girth → ∞) BJL ’17, BJP ’17

  • f random graphs with given degree sequence

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 5 / 18

slide-26
SLIDE 26

Greedy independence ratio – previous work

Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) Lauer & Wormald ’07 (same for d-regular graphs with girth → ∞) BJL ’17, BJP ’17 ι of random graphs with given degree sequence

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 5 / 18

slide-27
SLIDE 27

Random labelling

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-28
SLIDE 28

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-29
SLIDE 29

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-30
SLIDE 30

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-31
SLIDE 31

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-32
SLIDE 32

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.05 0.06 0.08 0.10 0.15 0.25 0.50 0.75 0.85 0.90 0.92 0.94 0.95

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-33
SLIDE 33

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87

➩

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-34
SLIDE 34

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87

➩

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-35
SLIDE 35

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87

➩

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-36
SLIDE 36

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87

➩

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-37
SLIDE 37

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87

➩

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-38
SLIDE 38

Random labelling

0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87

➩

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 6 / 18

slide-39
SLIDE 39

General framework

We wish to calculate the asymptotics of ι(Gn). We fjrst calculate I for chosen u.a.r. We hope that this is determined by a small neighbourhood of . This local view of is captured by the local limit of . 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

slide-40
SLIDE 40

General framework

We wish to calculate the asymptotics of ι(Gn). We fjrst calculate E(ι(Gn)) = P(ρn ∈ I(Gn)) for ρn chosen u.a.r. We hope that this is determined by a small neighbourhood of . This local view of is captured by the local limit of . 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

slide-41
SLIDE 41

General framework

We wish to calculate the asymptotics of ι(Gn). We fjrst calculate E(ι(Gn)) = P(ρn ∈ I(Gn)) for ρn chosen u.a.r. We hope that this is determined by a small neighbourhood of ρn. This local view of is captured by the local limit of . 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

slide-42
SLIDE 42

General framework

We wish to calculate the asymptotics of ι(Gn). We fjrst calculate E(ι(Gn)) = P(ρn ∈ I(Gn)) for ρn chosen u.a.r. We hope that this is determined by a small neighbourhood of ρn. This local view of ρn is captured by the local limit of Gn. 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

slide-43
SLIDE 43

General framework

We wish to calculate the asymptotics of ι(Gn). We fjrst calculate E(ι(Gn)) = P(ρn ∈ I(Gn)) for ρn chosen u.a.r. We hope that this is determined by a small neighbourhood of ρn. This local view of ρn is captured by the local limit of Gn. Decay of correlation = ⇒ ι(Gn) ∼ E(ι(Gn)) 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

slide-44
SLIDE 44

General framework

We wish to calculate the asymptotics of ι(Gn). We fjrst calculate E(ι(Gn)) = P(ρn ∈ I(Gn)) for ρn chosen u.a.r. We hope that this is determined by a small neighbourhood of ρn. This local view of ρn is captured by the local limit of Gn. Decay of correlation = ⇒ ι(Gn) ∼ E(ι(Gn)) 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

slide-45
SLIDE 45

Local limits

We say that a (random) graph sequence Gn locally converges to a random rooted graph (U, ρ), if for every r ≥ 0, the ball Br(G, ρn) converges in distribution to Br(U, ρ), where ρn is a uniform vertex of Gn.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18

slide-46
SLIDE 46

Local limits

We say that a (random) graph sequence Gn locally converges to a random rooted graph (U, ρ), if for every r ≥ 0, the ball Br(G, ρn) converges in distribution to Br(U, ρ), where ρn is a uniform vertex of Gn.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18

slide-47
SLIDE 47

Local limits

We say that a (random) graph sequence Gn locally converges to a random rooted graph (U, ρ), if for every r ≥ 0, the ball Br(G, ρn) converges in distribution to Br(U, ρ), where ρn is a uniform vertex of Gn.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18

slide-48
SLIDE 48

Local limits

We say that a (random) graph sequence Gn locally converges to a random rooted graph (U, ρ), if for every r ≥ 0, the ball Br(G, ρn) converges in distribution to Br(U, ρ), where ρn is a uniform vertex of Gn.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18

slide-49
SLIDE 49

Local limits

We say that a (random) graph sequence Gn locally converges to a random rooted graph (U, ρ), if for every r ≥ 0, the ball Br(G, ρn) converges in distribution to Br(U, ρ), where ρn is a uniform vertex of Gn.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18

slide-50
SLIDE 50

Local limits

We say that a (random) graph sequence Gn locally converges to a random rooted graph (U, ρ), if for every r ≥ 0, the ball Br(G, ρn) converges in distribution to Br(U, ρ), where ρn is a uniform vertex of Gn. · · ·

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18

slide-51
SLIDE 51

Local limits

We say that a (random) graph sequence Gn locally converges to a random rooted graph (U, ρ), if for every r ≥ 0, the ball Br(G, ρn) converges in distribution to Br(U, ρ), where ρn is a uniform vertex of Gn. · · ·

  • ➪

➪

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18

slide-52
SLIDE 52

Local limits

We say that a (random) graph sequence Gn locally converges to a random rooted graph (U, ρ), if for every r ≥ 0, the ball Br(G, ρn) converges in distribution to Br(U, ρ), where ρn is a uniform vertex of Gn. · · ·

  • ➪

➪

➪

➪

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18

slide-53
SLIDE 53

Local limits

We say that a (random) graph sequence Gn locally converges to a random rooted graph (U, ρ), if for every r ≥ 0, the ball Br(G, ρn) converges in distribution to Br(U, ρ), where ρn is a uniform vertex of Gn.

Examples

Pn, Cn

loc

− → Z [n]d loc − → Zd G(n, λ/n) loc − → Tλ, a Galton-Watson Pois(λ) tree Gn,d

loc

− → the d-regular tree Uniform random tree Tn

loc

− → ˆ T1, a size-biased GW Pois(1) tree Finite d-ary balanced tree loc − → the canopy tree

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 8 / 18

slide-54
SLIDE 54

Convergence of the greedy independence ratio

Theorem (Krivelevich, Mészáros, M., Shikhelman ’19+)

Suppose Gn has subfactorial growth. If Gn

loc

− → (U, ρ) then ι(Gn) → ι(U, ρ) a.a.s.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 9 / 18

slide-55
SLIDE 55

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-56
SLIDE 56

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-57
SLIDE 57

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-58
SLIDE 58

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-59
SLIDE 59

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-60
SLIDE 60

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-61
SLIDE 61

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-62
SLIDE 62

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-63
SLIDE 63

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-64
SLIDE 64

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-65
SLIDE 65

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-66
SLIDE 66

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-67
SLIDE 67

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-68
SLIDE 68

Decay of correlation

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 10 / 18

slide-69
SLIDE 69

Locally tree-like

We need to calculate ι(U, ρ), but even is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., 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

slide-70
SLIDE 70

Locally tree-like

We need to calculate ι(U, ρ), but even ι(Z2) is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., 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

slide-71
SLIDE 71

Locally tree-like

We need to calculate ι(U, ρ), but even ι(Z2) 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. Children of the past are roots to independent subtrees.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 11 / 18

slide-72
SLIDE 72

Locally tree-like

We need to calculate ι(U, ρ), but even ι(Z2) 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. ρ u1 u2 · · · ud Children of the past are roots to independent subtrees.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 11 / 18

slide-73
SLIDE 73

Locally tree-like

We need to calculate ι(U, ρ), but even ι(Z2) 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. ρ u1 u2 · · · ud Children of the past are roots to independent subtrees.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 11 / 18

slide-74
SLIDE 74

Systems of ordinary difgerential equations

Let (U, ρ) be a single-type branching process. I I I I

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18

slide-75
SLIDE 75

Systems of ordinary difgerential equations

Let (U, ρ) be a single-type branching process. y(x) = P(ρ ∈ I(U, ρ) ∧ σρ < x) I I I

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18

slide-76
SLIDE 76

Systems of ordinary difgerential equations

Let (U, ρ) be a single-type branching process. y(x) = P(ρ ∈ I(U, ρ) ∧ σρ < x) = x · P(ρ ∈ I(U, ρ)|σρ < x) = ∫ x P(ρ ∈ I(U, ρ)|σρ = z)dz I

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18

slide-77
SLIDE 77

Systems of ordinary difgerential equations

Let (U, ρ) be a single-type branching process. y(x) = P(ρ ∈ I(U, ρ) ∧ σρ < x) = x · P(ρ ∈ I(U, ρ)|σρ < x) = ∫ x P(ρ ∈ I(U, ρ)|σρ = z)dz y′(x) = P(ρ ∈ I(U, ρ)|σρ = x)

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18

slide-78
SLIDE 78

Systems of ordinary difgerential equations

Let (U, ρ) be a single-type branching process. y(x) = P(ρ ∈ I(U, ρ) ∧ σρ < x) = x · P(ρ ∈ I(U, ρ)|σρ < x) = ∫ x P(ρ ∈ I(U, ρ)|σρ = z)dz y′(x) = P(ρ ∈ I(U, ρ)|σρ = x) Thus, if y is a unique solution of y′(x) = ∑

ℓ∈N

P ( ξ<x = ℓ )( 1 − y(x) x )ℓ , y(0) = 0, then, ι(U, ρ) = y(1).

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18

slide-79
SLIDE 79

Systems of ordinary difgerential equations

Let (U, ρ) be a multi-type branching process. y(x) = P(ρ ∈ I(U, ρ) ∧ σρ < x) = x · P(ρ ∈ I(U, ρ)|σρ < x) = ∫ x P(ρ ∈ I(U, ρ)|σρ = z)dz y′(x) = P(ρ ∈ I(U, ρ)|σρ = x) Thus, if y is a unique solution of y′

k(x) =

∑

ℓ∈NT

∏

j∈T

P ( ξ<x

k→j = ℓj

)( 1 − yj(x) x )ℓj , yk(0) = 0, then, ι(U, ρ) = E(yk(1)).

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 12 / 18

slide-80
SLIDE 80

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-81
SLIDE 81

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-82
SLIDE 82

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-83
SLIDE 83

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-84
SLIDE 84

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-85
SLIDE 85

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-86
SLIDE 86

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-87
SLIDE 87

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-88
SLIDE 88

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-89
SLIDE 89

Size-biased Galton-Watson branching processes

Kolchin, Grimmett: the sequence of uniform random trees locally converges to the size-biased Galton-Watson Pois(1) tree.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 13 / 18

slide-90
SLIDE 90

Uniform random trees

y′

t(x) = ∞

∑

d=0

(λx)d eλxd! ( 1 − yt(x) x )d = e−λyt(x). hence yt(x) = ln(1 + λx)/λ. Thus ι(G(n, λ/n)) → ι(Tλ) = yt(1) = ln(1 + λ) λ .

s s t s

t

s

hence

s

, and for ,

s

, and we get

s

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 14 / 18

slide-91
SLIDE 91

Uniform random trees

y′

t(x) = ∞

∑

d=0

(λx)d eλxd! ( 1 − yt(x) x )d = e−λyt(x). hence yt(x) = ln(1 + λx)/λ. Thus ι(G(n, λ/n)) → ι(Tλ) = yt(1) = ln(1 + λ) λ . y′

s(x) = (1 − ys(x))y′ t(x) = (1 − ys(x))e−λyt(x) = 1 − ys(x)

1 + λx , hence ys(x) = 1 − (1 + λx)−1/λ, and for λ = 1, ys(1) = 1 − (1 + x)−1, and we get ι(Tn) → ι( ˆ T1) = ys(1) = 1 2.

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 14 / 18

slide-92
SLIDE 92

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

slide-93
SLIDE 93

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

slide-94
SLIDE 94

Simulations don’t lie (but I do)

red: 125 (50%), green: 92 (≈ 37%), blue: 32 (≈ 13%), black: 1

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 15 / 18

slide-95
SLIDE 95

Greedy independence ratio – results

Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) Lauer & Wormald ’07 (d-regular graphs with girth → ∞) KMMS ’19+ (same for functional digraphs)

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 16 / 18

slide-96
SLIDE 96

Greedy independence ratio – results

Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

✓

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) Lauer & Wormald ’07 (d-regular graphs with girth → ∞) KMMS ’19+ (same for functional digraphs)

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 16 / 18

slide-97
SLIDE 97

Greedy independence ratio – results

Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

✓

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ

✓

Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) Lauer & Wormald ’07 (d-regular graphs with girth → ∞) KMMS ’19+ (same for functional digraphs)

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 16 / 18

slide-98
SLIDE 98

Greedy independence ratio – results

Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

✓

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ

✓

Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) ✓ Lauer & Wormald ’07 (d-regular graphs with girth → ∞) KMMS ’19+ (same for functional digraphs)

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 16 / 18

slide-99
SLIDE 99

Greedy independence ratio – results

Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

✓

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ

✓

Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) ✓ Lauer & Wormald ’07 (d-regular graphs with girth → ∞) ✓ KMMS ’19+ (same for functional digraphs)

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 16 / 18

slide-100
SLIDE 100

Greedy independence ratio – results

Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

✓

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ

✓

Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) ✓ Lauer & Wormald ’07 (d-regular graphs with girth → ∞) ✓ KMMS ’19+ ι(Tn) → 1

2

✸

(same for functional digraphs)

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 16 / 18

slide-101
SLIDE 101

Greedy independence ratio – results

Flory ’39, Page ’59 ι(Pn) → 1

2(1 − e−2)

✓

McDiarmid ’84 ι(G(n, λ/n)) → ln(1 + λ)/λ

✓

Wormald ’95 ι(Gn,d) → 1

2

( 1 − (d − 1)−2/(d−2)) ✓ Lauer & Wormald ’07 (d-regular graphs with girth → ∞) ✓ KMMS ’19+ ι(Tn) → 1

2

✸

(same for functional digraphs)

✸

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 16 / 18

slide-102
SLIDE 102

Bonus: paths are the worst trees

Theorem (Krivelevich, Mészáros, M., Shikhelman ’19+)

If is a tree on vertices, then .

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 17 / 18

slide-103
SLIDE 103

Bonus: paths are the worst trees

ι(Pn) → 1

2

( 1 − e−2) ≈ 0.43233...

Theorem (Krivelevich, Mészáros, M., Shikhelman ’19+)

If is a tree on vertices, then .

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 17 / 18

slide-104
SLIDE 104

Bonus: paths are the worst trees

ι(Pn) → 1

2

( 1 − e−2) ≈ 0.43233... ι(Sn) → 1

Theorem (Krivelevich, Mészáros, M., Shikhelman ’19+)

If is a tree on vertices, then .

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 17 / 18

slide-105
SLIDE 105

Bonus: paths are the worst trees

ι(Pn) → 1

2

( 1 − e−2) ≈ 0.43233... ι(Sn) → 1

≤

Theorem (Krivelevich, Mészáros, M., Shikhelman ’19+)

If is a tree on vertices, then .

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 17 / 18

slide-106
SLIDE 106

Bonus: paths are the worst trees

ι(Pn) → 1

2

( 1 − e−2) ≈ 0.43233... ι(Sn) → 1

≤

Theorem (Krivelevich, Mészáros, M., Shikhelman ’19+)

If is a tree on vertices, then .

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 17 / 18

slide-107
SLIDE 107

Bonus: paths are the worst trees

ι(Pn) → 1

2

( 1 − e−2) ≈ 0.43233... ι(Sn) → 1

≤

Theorem (Krivelevich, Mészáros, M., Shikhelman ’19+)

If T is a tree on n vertices, then ι(Pn) ≤ ι(T).

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 17 / 18

slide-108
SLIDE 108

Thank You!

Peleg Michaeli (TAU) Greedy MIS July 21, 2019 18 / 18