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

β–Ά
distributed algorithms below the
SMART_READER_LITE
LIVE PREVIEW

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

Aug 14, 2022 348 likes β€’659 views

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 Unions Horizon 2020 Research and Innovation Programme under

slide-1
SLIDE 1

Distributed Algorithms Below the Greedy Regime

Yannic Maus

Coloring

This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement no. 755839.

Mohsen Ghaffari , Juho Hirvonen, Fabian Kuhn, Jara Uitto

slide-2
SLIDE 2

Communication Network = Problem Instance:

3

LOCAL Model [Linial; FOCS ’87]

5 15 6 11 1 21 33 26 9 27 8 2 7

Discrete synchronous rounds:

  • local computations
  • exchange messages with all neighbors

𝑯 = 𝑾, 𝑭 , 𝒐 = 𝑾

(computations unbounded, message sizes are unbounded)

time complexity = number of rounds

I am green

5 15 6 11 1 21 33 26 9 2 7 8 27 5 15 6 11 1 21 33 26 9 2 7 8 27

slide-3
SLIDE 3

Communication Network = Problem Instance:

4

CONGEST MODEL

5 15 6 11 1 21 33 26 9 27 8 2 7

Discrete synchronous rounds:

  • local computations
  • exchange messages with all neighbors

(computations unbounded, message sizes are unbounded) )

time complexity = number of rounds 𝑷(𝐦𝐩𝐑 𝒐) bits 𝑯 = 𝑾, 𝑭 , 𝒐 = 𝑾

5 15 6 11 1 21 33 26 9 2 7 8 27

slide-4
SLIDE 4

5

Classic Big Four (Greedy Regime)

Maximal Ind. Set (MIS) 𝚬 + 𝟐 -Vertex Coloring (πŸ‘πš¬ βˆ’ 𝟐)-Edge Coloring Maximal Matching

(Ξ”: maximum degree of 𝐻)

slide-5
SLIDE 5

6

In the LOCAL Model …

Greedy Below Greedy Maximal IS 2𝑃

log π‘œ

2𝑃

log π‘œ

Maximum IS, 𝟐 βˆ’ 𝝑 -approx. vertex cover, 2-approx. poly log π‘œ 2𝑃

log π‘œ

vertex cover, 𝟐 + 𝝑 -approx.

  • min. dominating set,

(𝟐 + 𝝑) 𝐦𝐩𝐑 𝚬-approx. 2𝑃

log π‘œ

2𝑃

log π‘œ

min dominating set, 𝟐 + 𝝑 -approx. hypergraph vertex cover, rank-approx. 2𝑃

log π‘œ

2𝑃

log π‘œ

hypergraph vertex cover, 𝟐 + 𝝑 -approx. 𝚬 + 𝟐 -vertex coloring 2𝑃

log π‘œ

2𝑃

log π‘œ

𝚬-vertex coloring πŸ‘πš¬ βˆ’ 𝟐 -edge coloring poly log π‘œ poly log π‘œ 𝟐 + 𝝑 𝚬-edge coloring maximal matching poly log π‘œ poly log π‘œ Maximum Matching, 𝟐 + 𝝑 -approx. CONGEST

slide-6
SLIDE 6

7

In the LOCAL Model …

Greedy Below Greedy Maximal IS 2𝑃

log π‘œ

2𝑃

log π‘œ

Maximum IS, 𝟐 βˆ’ 𝝑 -approx. 𝚬 + 𝟐 -vertex coloring 2𝑃

log π‘œ

2𝑃

log π‘œ

𝚬-vertex coloring πŸ‘πš¬ βˆ’ 𝟐 -edge coloring poly log π‘œ poly log π‘œ 𝟐 + 𝝑 𝚬-edge coloring maximal matching poly log π‘œ poly log π‘œ Maximum Matching, 𝟐 + 𝝑 -approx. CONGEST

β€œProblems that do not have easy sequential greedy algorithms.” β€œProblems that do have easy sequential greedy algorithms.”

slide-7
SLIDE 7

8

Outline

This Talk: How do we use LOCAL? Below Greedy Maximum IS 7/8-approx. Ξ©(π‘œ2) 2𝑃

log π‘œ

Maximum IS, 𝟐 βˆ’ 𝝑 -approx. vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

vertex cover, 𝟐 + 𝝑 -approx.

  • min. dominating set,

exact Ξ©(π‘œ2) 2𝑃

log π‘œ

min dominating set, 𝟐 + 𝝑 -approx. hypergraph vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

hypergraph vertex cover, 𝟐 + 𝝑 -approx. 𝝍(𝑯)-vertex coloring Ξ©(π‘œ2) 2𝑃

log π‘œ

𝚬-vertex coloring edge coloring ? poly log π‘œ 𝟐 + 𝝑 𝚬-edge coloring Maximum matching exact ? poly log π‘œ Maximum Matching, 𝟐 βˆ’ 𝝑 -approx.

Technique 1: Ball growing Technique 2: Local filling Technique 3: Aug. paths

CONGEST [Ghaffari, Kuhn, Maus; STOC ’17] [Ghaffari, Hirvonen, Kuhn, Maus; PODC ’18] [Ghaffari, Kuhn, Maus, Uitto; STOC ’18]

slide-8
SLIDE 8

9

Technique 1: Ball Growing

This Talk: How do we use LOCAL? Below Greedy Maximum IS 7/8-approx. Ξ©(π‘œ2) 2𝑃

log π‘œ

Maximum IS, 𝟐 βˆ’ 𝝑 -approx. vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

vertex cover, 𝟐 + 𝝑 -approx.

  • min. dominating set,

exact Ξ©(π‘œ2) 2𝑃

log π‘œ

min dominating set, 𝟐 + 𝝑 -approx. hypergraph vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

hypergraph vertex cover, 𝟐 + 𝝑 -approx. 𝝍(𝑯)-vertex coloring Ξ©(π‘œ2) 2𝑃

log π‘œ

𝚬-vertex coloring edge coloring ? poly log π‘œ 𝟐 + 𝝑 𝚬-edge coloring Maximum matching exact ? poly log π‘œ Maximum Matching, 𝟐 βˆ’ 𝝑 -approx.

Technique 1: Ball growing Technique 2: Local filling Technique 3: Aug. paths

slide-9
SLIDE 9

10

Sequential Ball Growing (MaxIS)

π’˜

Safe Ball π‘ͺ𝒔(π’˜): |𝑁𝑏𝑦𝐽𝑇(𝐢𝑠+1)| < (1 + πœ—) β‹… 𝑁𝑏𝑦𝐽𝑇(𝐢𝑠) Terminates with small radius 𝑠 = 𝑃(πœ—βˆ’1 log π‘œ). Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐢𝑠 is safe.

slide-10
SLIDE 10

11

Sequential Ball Growing (MaxIS)

Safe Ball π‘ͺ𝒔(π’˜): |𝑁𝑏𝑦𝐽𝑇(𝐢𝑠+1)| < (1 + πœ—) β‹… 𝑁𝑏𝑦𝐽𝑇(𝐢𝑠) Terminates with small radius 𝑠 = 𝑃(πœ—βˆ’1 log π‘œ). Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐢𝑠 is safe.

slide-11
SLIDE 11

12

Sequential Ball Growing (MaxIS)

Safe Ball π‘ͺ𝒔(π’˜): |𝑁𝑏𝑦𝐽𝑇(𝐢𝑠+1)| < (1 + πœ—) β‹… 𝑁𝑏𝑦𝐽𝑇(𝐢𝑠) Terminates with small radius 𝑠 = 𝑃(πœ—βˆ’1 log π‘œ). Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐢𝑠 is safe.

slide-12
SLIDE 12

13

Sequential Ball Growing (MaxIS)

Safe Ball π‘ͺ𝒔(π’˜): |𝑁𝑏𝑦𝐽𝑇(𝐢𝑠+1)| < (1 + πœ—) β‹… 𝑁𝑏𝑦𝐽𝑇(𝐢𝑠) Terminates with small radius 𝑠 = 𝑃(πœ—βˆ’1 log π‘œ). Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐢𝑠 is safe.

slide-13
SLIDE 13

14

Sequential Ball Growing (MaxIS)

Safe Ball π‘ͺ𝒔(π’˜): |𝑁𝑏𝑦𝐽𝑇(𝐢𝑠+1)| < (1 + πœ—) β‹… 𝑁𝑏𝑦𝐽𝑇(𝐢𝑠) Terminates with small radius 𝑠 = 𝑃(πœ—βˆ’1 log π‘œ). Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐢𝑠 is safe.

slide-14
SLIDE 14

15

Sequential Ball Growing (MaxIS)

Safe Ball π‘ͺ𝒔(π’˜): |𝑁𝑏𝑦𝐽𝑇(𝐢𝑠+1)| < (1 + πœ—) β‹… 𝑁𝑏𝑦𝐽𝑇(𝐢𝑠) Terminates with small radius 𝑠 = 𝑃(πœ—βˆ’1 log π‘œ). Find safe ball: Set 𝑠 = 0 and increase 𝑠 until ball 𝐢𝑠 is safe.

Sequentially computes a 1 + πœ— βˆ’1-approximation for MaxIS. (using unbounded computation)

slide-15
SLIDE 15

16

Parallel Ball Growing

Theorem Using (πͺ𝐩𝐦𝐳 𝐦𝐩𝐑 𝒐 , πͺ𝐩𝐦𝐳 𝐦𝐩𝐑 𝒐)-network decompositions β€œsequentially ball growing” can be β€œdone in parallel” in LOCAL. Corollary There are πͺ𝐩𝐦𝐳 𝐦𝐩𝐑 𝒐 randomized and πŸ‘π‘·

𝐦𝐩𝐑 𝒐 deterministic

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] [STOC ’17, Ghaffari, Kuhn, Maus]

slide-16
SLIDE 16

17

Technique 2: Local Filling

This Talk: How do we use LOCAL? Below Greedy Maximum IS 7/8-approx. Ξ©(π‘œ2) 2𝑃

log π‘œ

Maximum IS, 𝟐 βˆ’ 𝝑 -approx. vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

vertex cover, 𝟐 + 𝝑 -approx.

  • min. dominating set,

exact Ξ©(π‘œ2) 2𝑃

log π‘œ

min dominating set, 𝟐 + 𝝑 -approx. hypergraph vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

hypergraph vertex cover, 𝟐 + 𝝑 -approx. 𝝍(𝑯)-vertex coloring Ξ©(π‘œ2) 2𝑃

log π‘œ

𝚬-vertex coloring edge coloring ? poly log π‘œ 𝟐 + 𝝑 𝚬-edge coloring Maximum matching exact ? poly log π‘œ Maximum Matching, 𝟐 βˆ’ 𝝑 -approx.

Technique 1: Ball growing Technique 2: Local filling Technique 3: Aug. paths

slide-17
SLIDE 17

18

Ξ”-Coloring

Definition: An induced subgraph 𝐼 βŠ† 𝐻 is called an easy component if any 𝛦𝐻-coloring of 𝐻 βˆ– 𝐼 can be extended to a 𝛦𝐻-coloring of 𝐻 without changing the coloring on 𝐻 βˆ– 𝐼.

[PODC ’18; Ghaffari, Hirvonen, Kuhn, Maus]

Theorem: Let 𝐻 be a graph (β‰ clique) with max. degree Ξ” β‰₯ 3. Every node of 𝐻 has a small diameter easy component in distance at most O(log n). Well studied under the name degree chosable components.

β€œ β€œ

[ErdΕ‘s et al. ’79, Vizing β€˜76]

Previous Work: [Panconesi, Srinivasan; STOC ’93]

slide-18
SLIDE 18

19

Find an MIS 𝑡 of small diameter easy components Define 𝑃 log π‘œ Layers: 𝑴𝒋 = 𝑀 𝑀 in distance 𝒋 to some component in 𝑡} For 𝑗 = 𝑃(log π‘œ) to 1 color nodes in 𝑀𝑗 through solving a (deg+1)-list coloring Color easy components in M

slide-19
SLIDE 19

20

Find an MIS 𝑡 of small diameter easy components Define 𝑃 log π‘œ Layers: 𝑴𝒋 = 𝑀 𝑀 in distance 𝒋 to some component in 𝑡} For 𝑗 = 𝑃(log π‘œ) to 1 color nodes in 𝑀𝑗 through solving a (deg+1)-list coloring Color easy components in M

slide-20
SLIDE 20

21

Find an MIS 𝑡 of small diameter easy components Define 𝑃 log π‘œ Layers: 𝑴𝒋 = 𝑀 𝑀 in distance 𝒋 to some component in 𝑡} For 𝑗 = 𝑃(log π‘œ) to 1 color nodes in 𝑀𝑗 through solving a (deg+1)-list coloring Color easy components in M

Greedy regime

slide-21
SLIDE 21

22

Find an MIS 𝑡 of small diameter easy components Define 𝑃 log π‘œ Layers: 𝑴𝒋 = 𝑀 𝑀 in distance 𝒋 to some component in 𝑡} For 𝑗 = 𝑃(log π‘œ) to 1 color nodes in 𝑀𝑗 through solving a (deg+1)-list coloring Color easy components in M

Greedy regime

slide-22
SLIDE 22

23

Find an MIS 𝑡 of small diameter easy components Define 𝑃 log π‘œ Layers: 𝑴𝒋 = 𝑀 𝑀 in distance 𝒋 to some component in 𝑡} For 𝑗 = 𝑃(log π‘œ) to 1 color nodes in 𝑀𝑗 through solving a (deg+1)-list coloring Color easy components in M

Greedy regime

slide-23
SLIDE 23

24

Find an MIS 𝑡 of small diameter easy components Define 𝑃 log π‘œ Layers: 𝑴𝒋 = 𝑀 𝑀 in distance 𝒋 to some component in 𝑡} For 𝑗 = 𝑃(log π‘œ) to 1 color nodes in 𝑀𝑗 through solving a (deg+1)-list coloring Color easy components in M

Greedy regime

slide-24
SLIDE 24

25

Technique 3: Augmenting Paths

This Talk: How do we use LOCAL? Below Greedy Maximum IS 7/8-approx. Ξ©(π‘œ2) 2𝑃

log π‘œ

Maximum IS, 𝟐 βˆ’ 𝝑 -approx. vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

vertex cover, 𝟐 + 𝝑 -approx.

  • min. dominating set,

exact Ξ©(π‘œ2) 2𝑃

log π‘œ

min dominating set, 𝟐 + 𝝑 -approx. hypergraph vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

hypergraph vertex cover, 𝟐 + 𝝑 -approx. 𝝍(𝑯)-vertex coloring Ξ©(π‘œ2) 2𝑃

log π‘œ

𝚬-vertex coloring edge coloring ? poly log π‘œ 𝟐 + 𝝑 𝚬-edge coloring Maximum matching exact ? poly log π‘œ Maximum Matching, 𝟐 βˆ’ 𝝑 -approx.

Technique 1: Ball growing Technique 2: Local filling Technique 3: Aug. paths

slide-25
SLIDE 25

26

1 + πœ— Ξ”-Edge Coloring

Theorem 1 + πœ— Ξ”-edge coloring can be efficiently reduced to the computation of weighted maximum matching approximations . The reduction can be executed in the CONGEST model.

[Ghaffari, Kuhn, Maus, Uitto; STOC ’18]

For 𝑗 = 1 to 2Ξ” βˆ’ 1 compute a maximal matching 𝑁 of 𝐻 𝑁good color edges of 𝑁 with color 𝑗 𝑁good remove 𝑁 from 𝐻𝑁good Next Well known: 2Ξ” βˆ’ 1 iterations suffice to color all edges.

slide-26
SLIDE 26

27

1 + πœ— Ξ”-Edge Coloring

𝟐 + 𝝑 𝚬-edge coloring through good matchings: Reduce the max degree (amortized) at a rate of (1 βˆ’ πœ—). For 𝑗 = 1 to 1 + πœ— Ξ” compute a good matching 𝑁good of 𝐻 color edges of 𝑁good with color 𝑗 remove 𝑁good from 𝐻 Next

Theorem 1 + πœ— Ξ”-edge coloring can be efficiently reduced to the computation of weighted maximum matching approximations . The reduction can be executed in the CONGEST model.

[Ghaffari, Kuhn, Maus, Uitto; STOC ’18]

slide-27
SLIDE 27

Technique 1: Sequential ball growing

Problems: Approx. for MaxIS, MinDS, MinVC and many more … How do we (ab)use LOCAL? Compute optimal solutions in small diameter graphs

28

Summary Techniques (LOCAL)

Technique 2: Local filling

Problems: Ξ”-Coloring, ? How do we (ab)use LOCAL? β€œExistence” + small diameter is enough to obtain a solution

Technique 3: Augmenting paths

Problems: 1 + πœ— Ξ” Edge Coloring, Maximum Matching Approx. How do we (ab)use LOCAL? Finding a maximal set of augmenting paths

CONGEST

slide-28
SLIDE 28

29

Lower Bounds in CONGEST

Lower Bounds in CONGEST Below Greedy Maximum IS 7/8-approx. Ξ©(π‘œ2) 2𝑃

log π‘œ

Maximum IS, 𝟐 βˆ’ 𝝑 -approx. vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

vertex cover, 𝟐 + 𝝑 -approx.

  • min. dominating set,

exact Ξ©(π‘œ2) 2𝑃

log π‘œ

min dominating set, 𝟐 + 𝝑 -approx. hypergraph vertex cover, exact Ξ©(π‘œ2) 2𝑃

log π‘œ

hypergraph vertex cover, 𝟐 + 𝝑 -approx. 𝝍(𝑯)-vertex coloring Ξ©(π‘œ2) 2𝑃

log π‘œ

𝚬-vertex coloring ?-edge coloring ? poly log π‘œ 𝟐 + 𝝑 𝚬-edge coloring Maximum matching almost exact Ξ©( π‘œ poly log π‘œ Maximum Matching, 𝟐 βˆ’ 𝝑 -approx. CONGEST

[BCDELP β€˜19], [AKO ’18], [ACK ’16]

slide-29
SLIDE 29

30

Let’s Discuss …

A lot was spared in this talk (randomized!)

  • (1 βˆ’ πœ—) max cut approximation: [Zelke β€˜09]

Similar to technique 1: Subsampling + solving optimally

  • spanners, e.g., [Censor-Hillel, Dory; PODC β€˜18]
  • randomized edge coloring below the greedy regime, e.g.,

[Elkin, Pettie, Su; SODA ’15], [Chang, He, Li, Pettie, Uitto; SODA β€˜18], [Su, Vu; STOC β€˜19]

  • MPC: Maximum Matching approx. in time 𝑃(log log π‘œ)

[Behnezhad, Hajiaghayi, Harris; FOCS β€˜19]

  • lots more …

What can or cannot be done below the greedy regime in distributed models with limited communication (CONGEST, CONGESTED CLIQUE, MPC, …)?

Thank you