Three Challenges in Distributed Optimization Keren Censor-Hillel - PowerPoint PPT Presentation

Three Challenges in Distributed Optimization Keren Censor-Hillel Technion Workshop on Local Algorithms WOLA 2019 This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement no. 755839 1

Optimization problems 2

Optimization problems Minimum vertex cover 3

Optimization problems Minimum vertex cover NP-hard (2-ε)-approximation is UG-hard (some algs. with a smaller-than-2 apx) 4

Distributed optimization Minimum vertex cover Complexity ? 5

Distributed graph algorithms #Nodes = n #Bandwidth = B (typically O(logn)) Knowledge of neighbors CONGEST #Rounds = ? 6

Distributed graph algorithms #Nodes = n #Bandwidth = B (typically O(logn)) Knowledge of neighbors Models: CONGEST #Rounds = ? LOCAL 7

Distributed graph algorithms #Nodes = n #Bandwidth = B (typically O(logn)) Knowledge of neighbors Models: CONGEST #Rounds = ? LOCAL 8

Distributed graph algorithms #Nodes = n #Bandwidth = B (typically O(logn)) Knowledge of neighbors Models: 𝑃 𝑛 = 𝑃(𝑜 & ) CONGEST #Rounds = ? LOCAL 𝑃(𝐸) 9

Challenge 1: Distances 10

Challenge 1: Distances 11

Distances Exact minimum vertex cover: Ω(𝐸) rounds 12

Distances Exact minimum vertex cover: Ω(𝐸) rounds Approximations: LOCAL : (1 + 𝜗) -approximation in O(𝑞𝑝𝑚𝑧 (log 𝑜 /𝜗)) rounds [Ghaffari, Kuhn, Maus ‘17] 13

Distances Exact minimum vertex cover: Ω(𝐸) rounds Approximations: LOCAL : (1 + 𝜗) -approximation in O(𝑞𝑝𝑚𝑧 (log 𝑜 /𝜗)) rounds [Ghaffari, Kuhn, Maus ‘17] Ω(log Δ/ log log Δ) , Ω(√(log 𝑜 / log log 𝑜)) rounds [Kuhn, Moscibroda, Wattenhofer ‘04] Ω(1/𝜗) [Ben-Basat, Kawarabayashi, Schwartzman ‘18] 14

Optimal solutions for pieces of the graph LOCAL : (1 + 𝜗) -approximation in O(𝑞𝑝𝑚𝑧 (log 𝑜 /𝜗)) rounds [Ghaffari, Kuhn, Maus ‘17] Come to Yannic’s talk tomorrow! 15

Optimal solutions for pieces of the graph LOCAL : (1 + 𝜗) -approximation in O(𝑞𝑝𝑚𝑧 (log 𝑜 /𝜗)) rounds [Ghaffari, Kuhn, Maus ‘17] Come to Yannic’s talk tomorrow! 16

Optimal solutions for pieces of the graph LOCAL : (1 + 𝜗) -approximation in O(𝑞𝑝𝑚𝑧 (log 𝑜 /𝜗)) rounds [Ghaffari, Kuhn, Maus ‘17] Come to Yannic’s talk tomorrow! 17

Optimal solutions for pieces of the graph LOCAL : (1 + 𝜗) -approximation in O(𝑞𝑝𝑚𝑧 (log 𝑜 /𝜗)) rounds [Ghaffari, Kuhn, Maus ‘17] Come to Yannic’s talk tomorrow! 18

Optimal solutions for pieces of the graph LOCAL : (1 + 𝜗) -approximation in O(𝑞𝑝𝑚𝑧 (log 𝑜 /𝜗)) rounds [Ghaffari, Kuhn, Maus ‘17] Come to Yannic’s talk tomorrow! 19

Challenge 2: Congestion 20

Challenge 2: Congestion 21

Congestion LOCAL : (1 + 𝜗) -approximation in O(𝑞𝑝𝑚𝑧 (log 𝑜 /𝜗)) rounds [Ghaffari, Kuhn, Maus ‘17] Cannot collect dense neighborhoods quickly 22

CONGEST (2+ε)-approximation in 𝑃(log Δ/ log log Δ) rounds [Bar-Yehuda, C., Schwartzman ‘16] 2-approximation [Ben-Basat , Even, Kawarabayashi, Schwartzman ‘18] o(n 2 ), (2-ε)-approximation [Ben-Basat , Kawarabayashi, Schwartzman ‘18] Ω(log Δ/ log log Δ) , Ω(√(log 𝑜 / log log 𝑜)) rounds for approximation [Kuhn, Moscibroda, Wattenhofer ‘04] 23

CONGEST (2+ε)-approximation in 𝑃(log Δ/ log log Δ) rounds [Bar-Yehuda, C., Schwartzman ‘16] 2-approximation [Ben-Basat , Even, Kawarabayashi, Schwartzman ‘18] o(n 2 ), (2-ε)-approximation [Ben-Basat , Kawarabayashi, Schwartzman ‘18] Ω(log Δ/ log log Δ) , Ω(√(log 𝑜 / log log 𝑜)) rounds for approximation [Kuhn, Moscibroda, Wattenhofer ‘04] 24

CONGEST (2+ε)-approximation in 𝑃(log Δ/ log log Δ) rounds [Bar-Yehuda, C., Schwartzman ‘16] 2-approximation [Ben-Basat , Even, Kawarabayashi, Schwartzman ‘18] o(n 2 ), (2-ε)-approximation [Ben-Basat , Kawarabayashi, Schwartzman ‘18] Ω(log Δ/ log log Δ) , Ω(√(log 𝑜 / log log 𝑜)) rounds for approximation [Kuhn, Moscibroda, Wattenhofer ‘04] Ω(𝑜 & /𝑞𝑝𝑚𝑧 log 𝑜 ) rounds for exact [C., Khoury, Paz ‘17] 25

Minimum vertex cover in CONGEST #rounds 9 Ω(𝑜 & ) [C., Khoury, Paz ‘17] [Ben-Basat, Even, Kawarabayashi, Schwartzman ‘18] [Kuhn, Moscibroda, [Bar-Yehuda, C., approximation Wattenhofer ‘04] Schwartzman ‘16] 1 (exact) 2 2+ε 26

Minimum vertex cover in CONGEST #rounds 9 Ω(𝑜 & ) [C., Khoury, Paz ‘17] [Ben-Basat, Even, Kawarabayashi, Schwartzman ‘18] [Kuhn, Moscibroda, [Bar-Yehuda, C., approximation Wattenhofer ‘04] Schwartzman ‘16] 1 (exact) 2 2+ε 27

Minimum vertex cover in CONGEST [Bachrach, C., Dory, Efron, #rounds Leitersdorf, Paz ‘19] 9 Ω(𝑜 & ) [C., Khoury, Paz ‘17] [Ben-Basat, Even, Kawarabayashi, Schwartzman ‘18] [Kuhn, Moscibroda, [Bar-Yehuda, C., approximation Wattenhofer ‘04] Schwartzman ‘16] 1 (exact) 1.5 2 2+ε 28

Minimum vertex cover in CONGEST [Bachrach, C., Dory, Efron, #rounds Leitersdorf, Paz ‘19] 9 Ω(𝑜 & ) [C., Khoury, Paz ‘17] [Ben-Basat, Even, Kawarabayashi, Schwartzman ‘18] [Kuhn, Moscibroda, [Bar-Yehuda, C., approximation Wattenhofer ‘04] Schwartzman ‘16] 1 (exact) 1.5 2 2+ε 29

Minimum vertex cover in CONGEST [Bachrach, C., Dory, Efron, #rounds Leitersdorf, Paz ‘19] 9 Ω(𝑜 & ) [C., Khoury, Paz ‘17] [Ben-Basat, Even, Kawarabayashi, Schwartzman ‘18] [Kuhn, Moscibroda, [Bar-Yehuda, C., approximation Wattenhofer ‘04] Schwartzman ‘16] 1 (exact) 1.5 2 2+ε 30

2-Party communication [Yao ‘79] 31

History 32

History [Peleg, Rubinovich ‘97] [Lotker, Patt-Shamir, Peleg ’01] [Elkin ‘04] [Das-Sarma, Holzer, Kor, Korman, Nanongkai, Pandurangan, Peleg, Wattenhofer ‘11] [Frischknecht, Holzer, Wattenhofer ‘12] [Ghaffari, Kuhn ’13] [Drucker, Kuhn, Oshman ‘14] [Nanongkai, Das-Sarma, Pandurangan ‘14] [Das-Sarma, Molla, Pandurangan ‘15] [Holzer, Pinsker, ‘15] [Pandurangan, Peleg, Scquizzato ‘16] [Pandurangan, Robinson, Scquizzato ‘16] [C., Kavitha, Paz, Yehudayoff ’16] [Fischer, Gonen, Kuhn, Oshman ‘18] [C., Dory ‘17] [C., Dory ’18] 33

2-Party communication Alice: x=x 1 ,…,x k Bob: y=y 1 ,…,y k Goal: f(x,y) 34

2-Party communication Alice: x=x 1 ,…,x k Bob: y=y 1 ,…,y k Set-Disjointess: i: x i =y i =1 ? cost=Ω(k) [Kalyanasundaram and Schnitger ‘87] [Razborov ‘90] Goal: f(x,y) [Bar-Yossef et al. ‘04] 35

2-Party communication è CONGEST Alice: x=x 1 ,…,x k Bob: y=y 1 ,…,y k Graph property P of G x,y determines f(x,y) 36

2-Party communication è CONGEST Alice: x=x 1 ,…,x k Bob: y=y 1 ,…,y k Graph property P of G x,y determines f(x,y) Rounds Ÿ cut Ÿ B = Ω(cost(f(k))) 37

2-Party communication è CONGEST Alice: x=x 1 ,…,x k Bob: y=y 1 ,…,y k Graph property P of G x,y determines f(x,y) Rounds Ÿ cut Ÿ B = Ω(cost(f( k ))) Rounds = Ω(cost(f( k ))/ cut Ÿ B) 38

Warm-up: MVC, CONGEST Add edge iff 𝑜 : input is 0 cut = : 𝑜 = 𝑜/6 39

Warm-up: MVC, CONGEST Add edge iff 𝑜 : input is 0 cut = : 𝑜 = 𝑜/6 40

Warm-up: MVC, CONGEST Add edge iff 𝑜 : input is 0 cut = : 𝑜 = 𝑜/6 𝑜 & 𝑙 = : 41

Warm-up: MVC, CONGEST MVC = 4 : 𝑜 -2 inputs not disjoint 𝑜 : 42

Warm-up: MVC, CONGEST MVC = 4 : 𝑜 -2 inputs not disjoint 𝑜 : MVC ≥ 4 : 𝑜 -1 inputs disjoint 43

Warm-up: MVC, CONGEST MVC = 4 : 𝑜 -2 inputs not disjoint 𝑜 : MVC ≥ 4 : 𝑜 -1 inputs disjoint Lower bound: Ω(𝑙/𝑜 log 𝑜) = 9 Ω(𝑜) rounds 44

Minimum vertex cover in CONGEST [Bachrach, C., Dory, Efron, #rounds Leitersdorf, Paz ‘19] 9 Ω(𝑜 & ) [C., Khoury, Paz ‘17] [Ben-Basat, Even, Kawarabayashi, Schwartzman ‘18] [Kuhn, Moscibroda, [Bar-Yehuda, C., approximation Wattenhofer ‘04] Schwartzman ‘16] 1 (exact) 1.5 2 2+ε 45

Small cuts for MVC 0 1 0 1 [C., Khoury, Paz ‘17] Bit-gadget used for, e.g., diameter in [Abboud, C., Khoury ‘16] 46

Small cuts for MVC 0 1 0 1 [C., Khoury, Paz ‘17] MVC = 4( : 𝑜 -1+log : 𝑜 ) inputs not disjoint 47

Small cuts for MVC 0 1 0 1 [C., Khoury, Paz ‘17] k =Θ(n 2 ), cut =Θ(logn) Rounds = Ω(cost(f( k ))/ cut Ÿ logn) = Ω(n 2 /log 2 n) 48

Minimum vertex cover in CONGEST [Bachrach, C., Dory, Efron, #rounds Leitersdorf, Paz ‘19] 9 Ω(𝑜 & ) [C., Khoury, Paz ‘17] [Ben-Basat, Even, Kawarabayashi, Schwartzman ‘18] [Kuhn, Moscibroda, [Bar-Yehuda, C., approximation Wattenhofer ‘04] Schwartzman ‘16] 1 (exact) 1.5 2 2+ε 49

Why not for 1.5-approximation? • Alice/Bob compute OPT A and OPT B for their parts • The smaller is at most OPT/2 50

Why not for 1.5-approximation? • Alice/Bob compute OPT A and OPT B for their parts • The smaller is at most OPT/2 51

Why not for 1.5-approximation? • Alice/Bob compute OPT A and OPT B for their parts • The smaller is at most OPT/2 • The other side fills in the rest, at most OPT 52

Optimization problems 53