Distributed Algorithms Below the Greedy Regime Yannic Maus Mohsen - PowerPoint PPT Presentation

Coloring Distributed Algorithms Below the Greedy Regime Yannic Maus Mohsen Ghaffari , Juho Hirvonen, Fabian Kuhn, Jara Uitto This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement no. 755839.

LOCAL Model [Linial ; FOCS ’87] 𝑯 = 𝑾, 𝑭 , Communication Network = Problem Instance: 𝒐 = 𝑾 I am green 11 11 11 15 15 15 5 5 5 6 6 6 1 1 1 2 2 2 9 9 9 21 21 21 7 7 7 26 26 26 33 33 33 8 8 8 27 27 27 Discrete synchronous rounds: • local computations • exchange messages with all neighbors (computations unbounded, message sizes are unbounded) time complexity = number of rounds 3

CONGEST MODEL 𝑯 = 𝑾, 𝑭 , Communication Network = Problem Instance: 𝒐 = 𝑾 11 11 15 15 5 5 6 6 1 1 2 2 9 9 21 21 7 7 26 26 33 33 8 8 27 27 Discrete synchronous rounds: • local computations • exchange messages with all neighbors 𝑷(𝐦𝐩𝐡 𝒐) bits (computations unbounded, message sizes are unbounded) ) time complexity = number of rounds 4

Classic Big Four (Greedy Regime) 𝚬 + 𝟐 -Vertex Coloring Maximal Ind. Set (MIS) (𝟑𝚬 − 𝟐) -Edge Coloring Maximal Matching ( Δ : maximum degree of 𝐻 ) 5

In the LOCAL Model … Greedy Below Greedy 2 𝑃 log 𝑜 2 𝑃 log 𝑜 Maximal IS Maximum IS, 𝟐 − 𝝑 -approx. poly log 𝑜 vertex cover, 2 𝑃 log 𝑜 vertex cover, 𝟐 + 𝝑 -approx. 2 -approx. 2 𝑃 log 𝑜 2 𝑃 log 𝑜 min. dominating set, min dominating set, ( 𝟐 + 𝝑) 𝐦𝐩𝐡 𝚬 -approx. 𝟐 + 𝝑 -approx. 2 𝑃 log 𝑜 2 𝑃 log 𝑜 hypergraph vertex cover, hypergraph vertex cover, 𝟐 + 𝝑 -approx. rank -approx. 𝚬 + 𝟐 -vertex coloring 𝚬 -vertex coloring 2 𝑃 log 𝑜 2 𝑃 log 𝑜 𝟑𝚬 − 𝟐 -edge coloring poly log 𝑜 poly log 𝑜 𝟐 + 𝝑 𝚬 -edge coloring poly log 𝑜 poly log 𝑜 maximal matching Maximum Matching, 𝟐 + 𝝑 -approx. CONGEST 6

In the LOCAL Model … Greedy Below Greedy 2 𝑃 log 𝑜 2 𝑃 log 𝑜 Maximal IS Maximum IS, 𝟐 − 𝝑 -approx. “Problems that do have “ Problems that do not have easy sequential greedy easy sequential greedy algorithms .” algorithms .” 𝚬 + 𝟐 -vertex coloring 𝚬 -vertex coloring 2 𝑃 log 𝑜 2 𝑃 log 𝑜 𝟑𝚬 − 𝟐 -edge coloring poly log 𝑜 poly log 𝑜 𝟐 + 𝝑 𝚬 -edge coloring poly log 𝑜 poly log 𝑜 maximal matching Maximum Matching, 𝟐 + 𝝑 -approx. CONGEST 7

Outline This Talk: How do we use LOCAL? Below Greedy Ω(𝑜 2 ) 2 𝑃 log 𝑜 Maximum IS Technique 1: Ball growing Maximum IS, 𝟐 − 𝝑 -approx. 7/8 -approx. [Ghaffari, Kuhn, Maus; STOC ’ 17 ] Ω(𝑜 2 ) vertex cover, 2 𝑃 log 𝑜 vertex cover, 𝟐 + 𝝑 -approx. exact Ω(𝑜 2 ) 2 𝑃 log 𝑜 min. dominating set, min dominating set, 𝟐 + 𝝑 -approx. exact Ω(𝑜 2 ) 2 𝑃 log 𝑜 hypergraph vertex cover, hypergraph vertex cover, 𝟐 + 𝝑 -approx. exact Ω(𝑜 2 ) 𝝍(𝑯) -vertex coloring 𝚬 -vertex coloring Technique 2: Local filling 2 𝑃 log 𝑜 [Ghaffari, Hirvonen, Kuhn, Maus; PODC ’ 18 ] ? poly log 𝑜 𝟐 + 𝝑 𝚬 -edge coloring edge coloring Technique 3: Aug. paths ? poly log 𝑜 [Ghaffari, Kuhn, Maus , Uitto; STOC ’ 18 ] Maximum matching Maximum Matching, 𝟐 − 𝝑 -approx. exact CONGEST 8

Technique 1: Ball Growing This Talk: How do we use LOCAL? Below Greedy Ω(𝑜 2 ) 2 𝑃 log 𝑜 Maximum IS Technique 1: Ball growing Maximum IS, 𝟐 − 𝝑 -approx. 7/8 -approx. Ω(𝑜 2 ) vertex cover, 2 𝑃 log 𝑜 vertex cover, 𝟐 + 𝝑 -approx. exact Ω(𝑜 2 ) 2 𝑃 log 𝑜 min. dominating set, min dominating set, 𝟐 + 𝝑 -approx. exact Ω(𝑜 2 ) 2 𝑃 log 𝑜 hypergraph vertex cover, hypergraph vertex cover, 𝟐 + 𝝑 -approx. exact Ω(𝑜 2 ) 𝝍(𝑯) -vertex coloring 𝚬 -vertex coloring Technique 2: Local filling 2 𝑃 log 𝑜 ? poly log 𝑜 𝟐 + 𝝑 𝚬 -edge coloring edge coloring Technique 3: Aug. paths ? poly log 𝑜 Maximum matching Maximum Matching, 𝟐 − 𝝑 -approx. exact 9

Sequential Ball Growing (MaxIS) 𝒘 Safe Ball 𝑪 𝒔 (𝒘) : |𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠+1 )| < (1 + 𝜗) ⋅ 𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠 ) Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐶 𝑠 is safe. Terminates with small radius 𝑠 = 𝑃(𝜗 −1 log 𝑜) . 10

Sequential Ball Growing (MaxIS) Safe Ball 𝑪 𝒔 (𝒘) : |𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠+1 )| < (1 + 𝜗) ⋅ 𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠 ) Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐶 𝑠 is safe. Terminates with small radius 𝑠 = 𝑃(𝜗 −1 log 𝑜) . 11

Sequential Ball Growing (MaxIS) Safe Ball 𝑪 𝒔 (𝒘) : |𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠+1 )| < (1 + 𝜗) ⋅ 𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠 ) Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐶 𝑠 is safe. Terminates with small radius 𝑠 = 𝑃(𝜗 −1 log 𝑜) . 12

Sequential Ball Growing (MaxIS) Safe Ball 𝑪 𝒔 (𝒘) : |𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠+1 )| < (1 + 𝜗) ⋅ 𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠 ) Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐶 𝑠 is safe. Terminates with small radius 𝑠 = 𝑃(𝜗 −1 log 𝑜) . 13

Sequential Ball Growing (MaxIS) Safe Ball 𝑪 𝒔 (𝒘) : |𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠+1 )| < (1 + 𝜗) ⋅ 𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠 ) Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐶 𝑠 is safe. Terminates with small radius 𝑠 = 𝑃(𝜗 −1 log 𝑜) . 14

Sequential Ball Growing (MaxIS) Sequentially computes a 1 + 𝜗 −1 -approximation for MaxIS. (using unbounded computation) Safe Ball 𝑪 𝒔 (𝒘) : |𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠+1 )| < (1 + 𝜗) ⋅ 𝑁𝑏𝑦𝐽𝑇(𝐶 𝑠 ) Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐶 𝑠 is safe. Terminates with small radius 𝑠 = 𝑃(𝜗 −1 log 𝑜) . 15

Parallel Ball Growing Theorem Using ( 𝐪𝐩𝐦𝐳 𝐦𝐩𝐡 𝒐 , 𝐪𝐩𝐦𝐳 𝐦𝐩𝐡 𝒐) -network decompositions “sequentially ball growing” can be “done in parallel” in LOCAL. [ STOC ’ 17, Ghaffari, Kuhn, Maus ] Corollary 𝐦𝐩𝐡 𝒐 deterministic There are 𝐪𝐩𝐦𝐳 𝐦𝐩𝐡 𝒐 randomized and 𝟑 𝑷 𝟐 + 𝝑 -approximation algorithms for covering and packing integer linear programs . This includes maximum independent set, minimum dominating set, vertex cover, … . [ STOC ’ 17, Ghaffari, Kuhn, Maus ] 16

Technique 2: Local Filling This Talk: How do we use LOCAL? Below Greedy Ω(𝑜 2 ) 2 𝑃 log 𝑜 Maximum IS Technique 1: Ball growing Maximum IS, 𝟐 − 𝝑 -approx. 7/8 -approx. Ω(𝑜 2 ) vertex cover, 2 𝑃 log 𝑜 vertex cover, 𝟐 + 𝝑 -approx. exact Ω(𝑜 2 ) 2 𝑃 log 𝑜 min. dominating set, min dominating set, 𝟐 + 𝝑 -approx. exact Ω(𝑜 2 ) 2 𝑃 log 𝑜 hypergraph vertex cover, hypergraph vertex cover, 𝟐 + 𝝑 -approx. exact Ω(𝑜 2 ) 𝝍(𝑯) -vertex coloring 𝚬 -vertex coloring Technique 2: Local filling 2 𝑃 log 𝑜 ? poly log 𝑜 𝟐 + 𝝑 𝚬 -edge coloring edge coloring Technique 3: Aug. paths ? poly log 𝑜 Maximum matching Maximum Matching, 𝟐 − 𝝑 -approx. exact 17

Δ -Coloring Previous Work: [Panconesi, Srinivasan; STOC ’93 ] Definition: An induced subgraph 𝐼 ⊆ 𝐻 is called an easy component if any 𝛦 𝐻 -coloring of 𝐻 ∖ 𝐼 can be extended to a 𝛦 𝐻 -coloring of 𝐻 without changing the coloring on 𝐻 ∖ 𝐼 . Well studied under the name degree chosable components. [ Erdős et al. ’79, Vizing ‘76 ] “ “ Theorem: Let 𝐻 be a graph ( ≠ clique) with max. degree Δ ≥ 3 . Every node of 𝐻 has a small diameter easy component in distance at most O(log n) . [ PODC ’ 18; Ghaffari, Hirvonen, Kuhn, Maus ] 18

Find an MIS 𝑵 of small diameter easy components Define 𝑃 log 𝑜 Layers: 𝑴 𝒋 = 𝑤 𝑤 in distance 𝒋 to some component in 𝑵} For 𝑗 = 𝑃(log 𝑜) to 1 color nodes in 𝑀 𝑗 through solving a (deg+1)-list coloring Color easy components in M 19

Find an MIS 𝑵 of small diameter easy components Define 𝑃 log 𝑜 Layers: 𝑴 𝒋 = 𝑤 𝑤 in distance 𝒋 to some component in 𝑵} For 𝑗 = 𝑃(log 𝑜) to 1 color nodes in 𝑀 𝑗 through solving a (deg+1)-list coloring Color easy components in M 20

Find an MIS 𝑵 of small diameter easy components Greedy regime Define 𝑃 log 𝑜 Layers: 𝑴 𝒋 = 𝑤 𝑤 in distance 𝒋 to some component in 𝑵} For 𝑗 = 𝑃(log 𝑜) to 1 color nodes in 𝑀 𝑗 through solving a (deg+1)-list coloring Color easy components in M 21

Find an MIS 𝑵 of small diameter easy components Greedy regime Define 𝑃 log 𝑜 Layers: 𝑴 𝒋 = 𝑤 𝑤 in distance 𝒋 to some component in 𝑵} For 𝑗 = 𝑃(log 𝑜) to 1 color nodes in 𝑀 𝑗 through solving a (deg+1)-list coloring Color easy components in M 22

Find an MIS 𝑵 of small diameter easy components Greedy regime Define 𝑃 log 𝑜 Layers: 𝑴 𝒋 = 𝑤 𝑤 in distance 𝒋 to some component in 𝑵} For 𝑗 = 𝑃(log 𝑜) to 1 color nodes in 𝑀 𝑗 through solving a (deg+1)-list coloring Color easy components in M 23