Vertex Sparsification and Oblivious Reductions Ankur Moitra, MIT - - PowerPoint PPT Presentation

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Vertex Sparsification and Oblivious Reductions Ankur Moitra, MIT - - PowerPoint PPT Presentation

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Vertex Sparsification and Oblivious Reductions Ankur Moitra, MIT September 14, 2010 Ankur Moitra (MIT) Sparsification September 14, 2010 Background The Minimum Bisection Problem Goal: Minimize cost of bisection Ankur Moitra (MIT)

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SLIDE 1

Vertex Sparsification and Oblivious Reductions

Ankur Moitra, MIT September 14, 2010

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 2

Background

The Minimum Bisection Problem

Goal: Minimize cost of bisection

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 3

Background

The Minimum Bisection Problem

Goal: Minimize cost of bisection

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 4

Background

The Minimum Bisection Problem

4 Goal: Minimize cost of bisection cost =

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 5

Background

History of the Minimum Bisection Problem

1 Applications through Divide-and-Conquer: VLSI design, sparse matrix

computations, approximation algorithms

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 6

Background

History of the Minimum Bisection Problem

1 Applications through Divide-and-Conquer: VLSI design, sparse matrix

computations, approximation algorithms

2 [Kernighan, Lin 1970] Local search heuristic Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 7

Background

History of the Minimum Bisection Problem

1 Applications through Divide-and-Conquer: VLSI design, sparse matrix

computations, approximation algorithms

2 [Kernighan, Lin 1970] Local search heuristic 3 [Garey, Johnson, Stockmeyer 1976] NP-Complete Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 8

Background

History of the Minimum Bisection Problem

1 Applications through Divide-and-Conquer: VLSI design, sparse matrix

computations, approximation algorithms

2 [Kernighan, Lin 1970] Local search heuristic 3 [Garey, Johnson, Stockmeyer 1976] NP-Complete 4 [Leighton, Rao 1988] O(log n) approximate minimum bisection Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 9

Background

History of the Minimum Bisection Problem

1 Applications through Divide-and-Conquer: VLSI design, sparse matrix

computations, approximation algorithms

2 [Kernighan, Lin 1970] Local search heuristic 3 [Garey, Johnson, Stockmeyer 1976] NP-Complete 4 [Leighton, Rao 1988] O(log n) approximate minimum bisection 5 [Saran, Vazirani 1995] n

2-approximation algorithm

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 10

Background

History of the Minimum Bisection Problem

1 Applications through Divide-and-Conquer: VLSI design, sparse matrix

computations, approximation algorithms

2 [Kernighan, Lin 1970] Local search heuristic 3 [Garey, Johnson, Stockmeyer 1976] NP-Complete 4 [Leighton, Rao 1988] O(log n) approximate minimum bisection 5 [Saran, Vazirani 1995] n

2-approximation algorithm

6 [Arora, Karger, Karpinski 1999] PTAS for dense graphs Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 11

Background

History of the Minimum Bisection Problem

1 Applications through Divide-and-Conquer: VLSI design, sparse matrix

computations, approximation algorithms

2 [Kernighan, Lin 1970] Local search heuristic 3 [Garey, Johnson, Stockmeyer 1976] NP-Complete 4 [Leighton, Rao 1988] O(log n) approximate minimum bisection 5 [Saran, Vazirani 1995] n

2-approximation algorithm

6 [Arora, Karger, Karpinski 1999] PTAS for dense graphs 7 [Feige, Krauthgamer 2001] O(log1.5 n)-approximation algorithm Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 12

Background

History of the Minimum Bisection Problem

1 Applications through Divide-and-Conquer: VLSI design, sparse matrix

computations, approximation algorithms

2 [Kernighan, Lin 1970] Local search heuristic 3 [Garey, Johnson, Stockmeyer 1976] NP-Complete 4 [Leighton, Rao 1988] O(log n) approximate minimum bisection 5 [Saran, Vazirani 1995] n

2-approximation algorithm

6 [Arora, Karger, Karpinski 1999] PTAS for dense graphs 7 [Feige, Krauthgamer 2001] O(log1.5 n)-approximation algorithm 8 [Khot 2004] No PTAS, unless P = NP Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 13

Background

History of the Minimum Bisection Problem

1 Applications through Divide-and-Conquer: VLSI design, sparse matrix

computations, approximation algorithms

2 [Kernighan, Lin 1970] Local search heuristic 3 [Garey, Johnson, Stockmeyer 1976] NP-Complete 4 [Leighton, Rao 1988] O(log n) approximate minimum bisection 5 [Saran, Vazirani 1995] n

2-approximation algorithm

6 [Arora, Karger, Karpinski 1999] PTAS for dense graphs 7 [Feige, Krauthgamer 2001] O(log1.5 n)-approximation algorithm 8 [Khot 2004] No PTAS, unless P = NP 9 [R¨

acke, 2008] O(log n)-approximation algorithm

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 14

Background

The Steiner Minimum Bisection Problem

Goal: Minimize cost of a bisection of the k blue nodes

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 15

Background

The Steiner Minimum Bisection Problem

Goal: Minimize cost of a bisection of the k blue nodes

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 16

Background

The Steiner Minimum Bisection Problem

4 Goal: Minimize cost of a bisection of the k blue nodes cost =

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 17

Background

Question Can we find a poly(log k)-approximation algorithm?

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 18

Background

Question Can we find a poly(log k)-approximation algorithm? Some approximation guarantees can be made independent of the graph size:

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 19

Background

Question Can we find a poly(log k)-approximation algorithm? Some approximation guarantees can be made independent of the graph size:

1 O(log k) generalized sparsest cut for k commodities

[Linial, London, Rabinovich 1995] and [Aumann, Rabani 1997]

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 20

Background

Question Can we find a poly(log k)-approximation algorithm? Some approximation guarantees can be made independent of the graph size:

1 O(log k) generalized sparsest cut for k commodities

[Linial, London, Rabinovich 1995] and [Aumann, Rabani 1997]

2 O(log k) multicut for k terminals

[Garg, Vazirani, Yannakakis 1996]

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 21

Background

Question Can we find a poly(log k)-approximation algorithm? Some approximation guarantees can be made independent of the graph size:

1 O(log k) generalized sparsest cut for k commodities

[Linial, London, Rabinovich 1995] and [Aumann, Rabani 1997]

2 O(log k) multicut for k terminals

[Garg, Vazirani, Yannakakis 1996]

3 O(

log k log log k ) 0-extension for k terminals

[Fakcharoenphol, Harrelson, Rao, Talwar 2003]

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 22

Background

A Meta Question

Given A poly(log n) approximation algorithm (integrality gap or competitive ratio) for an optimization problem characterized by cuts or flows

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 23

Background

A Meta Question

Given A poly(log n) approximation algorithm (integrality gap or competitive ratio) for an optimization problem characterized by cuts or flows Let k be the number of ”interesting” nodes

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 24

Background

A Meta Question

Given A poly(log n) approximation algorithm (integrality gap or competitive ratio) for an optimization problem characterized by cuts or flows Let k be the number of ”interesting” nodes Meta Question Can we give a poly(log k) approximation algorithm (integrality gap or competitive ratio)?

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 25

Background

A Meta Question

Given A poly(log n) approximation algorithm (integrality gap or competitive ratio) for an optimization problem characterized by cuts or flows Let k be the number of ”interesting” nodes Meta Question Can we give a poly(log k) approximation algorithm (integrality gap or competitive ratio)? Yes we can...

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 26

Background

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for:

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 27

Background

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for: Steiner minimum bisection,

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 28

Background

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for: Steiner minimum bisection, requirement cut,

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 29

Background

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for: Steiner minimum bisection, requirement cut,

l-multicut,

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 30

Background

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for: Steiner minimum bisection, requirement cut,

l-multicut, oblivious 0-extension,

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 31

Background

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for: Steiner minimum bisection, requirement cut,

l-multicut, oblivious 0-extension, and Steiner generalizations of oblivious routing,

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 32

Background

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for: Steiner minimum bisection, requirement cut,

l-multicut, oblivious 0-extension, and Steiner generalizations of oblivious routing, min-cut linear arrangement,

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 33

Background

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for: Steiner minimum bisection, requirement cut,

l-multicut, oblivious 0-extension, and Steiner generalizations of oblivious routing, min-cut linear arrangement, and minimum linear arrangement

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 34

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a b d a c d b Graph G=(V,E) Sparsifier G’=(K,E’)

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 35

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a b d a c d b Graph G=(V,E) Sparsifier G’=(K,E’)

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 36

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a b d a c d b Graph G=(V,E) Sparsifier G’=(K,E’)

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 37

Vertex Sparsifiers

General Approach: Cut Sparsifiers

h (a) = 5 d

K

a c d b Graph G=(V,E) Sparsifier G’=(K,E’) c a b

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 38

Vertex Sparsifiers

General Approach: Cut Sparsifiers

h (a) = 5 d

K

a c d b Graph G=(V,E) Sparsifier G’=(K,E’) c a b

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 39

Vertex Sparsifiers

General Approach: Cut Sparsifiers

h (a) = 5 d

K

a c d b Graph G=(V,E) Sparsifier G’=(K,E’) c a b

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 40

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 41

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 42

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 43

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K

h (b) = 2

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 44

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K

h (b) = 2

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 45

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K

h (b) = 2

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 46

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 47

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 48

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 49

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K

h (ac) = 4

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 50

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K

h (ac) = 4

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 51

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c a c d b Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K

h (ac) = 4

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 52

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c 2 1 __ 2 Graph G=(V,E) Sparsifier G’=(K,E’) a b d a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 53

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c 1 __ 2 h (a) = 5 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K

a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 54

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c __ 2 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K

a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 55

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c __ 2 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K

a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 56

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c __ 2 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K

h (b) = 2 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 57

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c 2 h’(b) = 2.5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K

h (b) = 2 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 58

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c 2 h’(b) = 2.5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K

h (b) = 2 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 59

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c 2 h’(b) = 2.5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K

h (b) = 2 h (ac) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 60

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c h’(b) = 2.5 h’(ac) = 4.5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K

h (b) = 2 h (ac) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 61

Vertex Sparsifiers

General Approach: Cut Sparsifiers

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 62

Vertex Sparsifiers

General Approach: Cut Sparsifiers

5 h (a) = 5 h’(a) = 6 c h (b) = 2

Quality = ___

4 a b d

K K K K K K K

h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5 h’(ad) = 5 h’(d) = 5

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 63

Vertex Sparsifiers

Cut Sparsifiers, Informally

Definition G ′ = (K, E ′) is a Cut Sparsifier for G = (V , E) if all cuts in G ′ are at least as large as the corresponding min-cut in G.

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 64

Vertex Sparsifiers

Cut Sparsifiers, Informally

Definition G ′ = (K, E ′) is a Cut Sparsifier for G = (V , E) if all cuts in G ′ are at least as large as the corresponding min-cut in G. Definition The Quality of a Cut Sparsifier is the maximum ratio of a cut in G ′ to the corresponding min-cut in G.

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 65

Vertex Sparsifiers

Cut Sparsifiers

Good quality Cut Sparsifiers exist!

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 66

Vertex Sparsifiers

Cut Sparsifiers

Good quality Cut Sparsifiers exist! And such graphs can be computed efficiently!

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 67

Vertex Sparsifiers

Cut Sparsifiers

Good quality Cut Sparsifiers exist! And such graphs can be computed efficiently! Theorem (Moitra, FOCS 2009) For all (undirected) weighted graphs G = (V , E), and all K ⊂ V there is an (undirected) weighted graph G ′ = (K, E ′) such that G ′ is a O(log k/ log log k)-quality Cut Sparsifier.

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 68

Vertex Sparsifiers

Cut Sparsifiers

Good quality Cut Sparsifiers exist! And such graphs can be computed efficiently! Theorem (Moitra, FOCS 2009) For all (undirected) weighted graphs G = (V , E), and all K ⊂ V there is an (undirected) weighted graph G ′ = (K, E ′) such that G ′ is a O(log k/ log log k)-quality Cut Sparsifier. This bound improves to O(1) if G is planar, or if G excludes any fixed minor!

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 69

Vertex Sparsifiers

An Application to Steiner Minimum Bisection

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 70

Vertex Sparsifiers

An Application to Steiner Minimum Bisection

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-71
SLIDE 71

Vertex Sparsifiers

An Application to Steiner Minimum Bisection

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-72
SLIDE 72

Vertex Sparsifiers

An Application to Steiner Minimum Bisection

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-73
SLIDE 73

Vertex Sparsifiers

An Application to Steiner Minimum Bisection

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-74
SLIDE 74

Vertex Sparsifiers

An Application to Steiner Minimum Bisection

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-75
SLIDE 75

Vertex Sparsifiers

An Application to Steiner Minimum Bisection

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-76
SLIDE 76

Vertex Sparsifiers

An Application to Steiner Minimum Bisection

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 77

Vertex Sparsifiers

An Application to Steiner Minimum Bisection

c h’(ad) = 5 h’(d) = 5 h (a) = 5 h’(a) = 6 Graph G=(V,E) Sparsifier G’=(K,E’) a b d

K K K K K K K

h (b) = 2 h (c) = 3 h (ab) = 7 h (ad) = 5 h (ac) = 4 h (d) = 4 a c d b 3 __ 2 3 __ 2 5 __ 2 1 1 __ 2 1 __ 2 h’(b) = 2.5 h’(c) = 3.5 h’(ab) = 7.5 h’(ac) = 4.5

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 78

Vertex Sparsifiers

This is a general strategy!

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 79

Vertex Sparsifiers

This is a general strategy! For any problem characterized by cuts or flows:

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 80

Vertex Sparsifiers

This is a general strategy! For any problem characterized by cuts or flows:

1 Construct G ′ so OPT ′ ≤ poly(log k)OPT Ankur Moitra (MIT) Sparsification September 14, 2010

slide-81
SLIDE 81

Vertex Sparsifiers

This is a general strategy! For any problem characterized by cuts or flows:

1 Construct G ′ so OPT ′ ≤ poly(log k)OPT 2 Run approximation algorithm on G ′ Ankur Moitra (MIT) Sparsification September 14, 2010

slide-82
SLIDE 82

Vertex Sparsifiers

This is a general strategy! For any problem characterized by cuts or flows:

1 Construct G ′ so OPT ′ ≤ poly(log k)OPT 2 Run approximation algorithm on G ′ 3 Map solution back to G Ankur Moitra (MIT) Sparsification September 14, 2010

slide-83
SLIDE 83

Vertex Sparsifiers

This is a general strategy! For any problem characterized by cuts or flows:

1 Construct G ′ so OPT ′ ≤ poly(log k)OPT 2 Run approximation algorithm on G ′ 3 Map solution back to G

This will bootstrap a poly(log k) guarantee from a poly(log n) guarantee

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 84

Vertex Sparsifiers

Oblivious Reductions

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 85

Vertex Sparsifiers

Oblivious Reductions

This approach is useful even for efficiently solvable problems!

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-86
SLIDE 86

Vertex Sparsifiers

Oblivious Reductions

This approach is useful even for efficiently solvable problems! Question What if we are asked to solve a routing problem on K, but we don’t yet know the demands?

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-87
SLIDE 87

Vertex Sparsifiers

Oblivious Reductions

This approach is useful even for efficiently solvable problems! Question What if we are asked to solve a routing problem on K, but we don’t yet know the demands?

1 Construct G ′ Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 88

Vertex Sparsifiers

Oblivious Reductions

This approach is useful even for efficiently solvable problems! Question What if we are asked to solve a routing problem on K, but we don’t yet know the demands?

1 Construct G ′

Given G ′, there will be a canonical way to map flows in G ′ back to G

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-89
SLIDE 89

Vertex Sparsifiers

Oblivious Reductions

This approach is useful even for efficiently solvable problems! Question What if we are asked to solve a routing problem on K, but we don’t yet know the demands?

1 Construct G ′

Given G ′, there will be a canonical way to map flows in G ′ back to G

2 Given demands, optimally solve on G ′ Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 90

Vertex Sparsifiers

Oblivious Reductions

This approach is useful even for efficiently solvable problems! Question What if we are asked to solve a routing problem on K, but we don’t yet know the demands?

1 Construct G ′

Given G ′, there will be a canonical way to map flows in G ′ back to G

2 Given demands, optimally solve on G ′ 3 Map solution back to G Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 91

Vertex Sparsifiers

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for: Steiner minimum bisection, requirement cut,

l-multicut, oblivious 0-extension, and Steiner generalizations of oblivious routing, min-cut linear arrangement, and minimum linear arrangement

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 92

Vertex Sparsifiers

Highlights

1 Approximation Guarantees Independent of the Graph Size:

We give the first poly(log k) approximation algorithms (or competitive ratios) for: Steiner minimum bisection, requirement cut,

l-multicut, oblivious 0-extension, and Steiner generalizations of oblivious routing, min-cut linear arrangement, and minimum linear arrangement

2 Oblivious Reductions: All you need to know about the underlying

communication network is its vertex sparsifier

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 93

Vertex Sparsifiers

Definition

G=(V,E)

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SLIDE 94

Vertex Sparsifiers

Definition

G=(V,E)

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SLIDE 95

Vertex Sparsifiers

Definition

G =(K,E )

f f

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SLIDE 96

Vertex Sparsifiers

Definition Let f : V → K, is a 0-extension if for all a ∈ K, f (a) = a.

G =(K,E )

f f

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SLIDE 97

Vertex Sparsifiers

Lemma Gf is a Cut Sparsifier

G =(K,E )

f f

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 98

Vertex Sparsifiers

Lemma Gf is a Cut Sparsifier

G =(K,E ) K−A A

f f

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 99

Vertex Sparsifiers

Lemma Gf is a Cut Sparsifier

G =(K,E ) K−A A

f f

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 100

Vertex Sparsifiers

Lemma Gf is a Cut Sparsifier

A G=(V,E) K−A

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SLIDE 101

Vertex Sparsifiers

Lemma Gf is a Cut Sparsifier

A G=(V,E) K−A

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 102

Vertex Sparsifiers

Lemma Gf is a Cut Sparsifier

A G=(V,E) K−A

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 103

Vertex Sparsifiers

Lemma Gf is a Cut Sparsifier

A G=(V,E) K−A

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 104

An Example

An Example

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slide-105
SLIDE 105

An Example

An Example

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-106
SLIDE 106

An Example

An Example

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-107
SLIDE 107

An Example

An Example

k−1 The cost is

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 108

An Example

An Example

k−1 The cost is The cost is 1

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 109

An Example

An Example

= k−1 The cost is The cost is 1 k−1 Quality

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 110

An Example

An Example: A Second Attempt

k p = 1 __ k p = 1 __

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 111

An Example

An Example: A Second Attempt

k __ k 1 __ k 1 __ k 1 k __ 1 __ p = 1 __ k p = 1 __ k 1

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 112

An Example

An Example: A Second Attempt

k k 2 1 __ k 1 __ k 1 __ k 1 __ p = 1 __ k p = 1 __ k __ k 1 __ k 1 __ k 1 k __ 1 __

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 113

An Example

An Example: A Second Attempt

k p = 1 __ k p = 1 __ k 2 __

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 114

An Example

An Example: A Second Attempt

k 2 __

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 115

An Example

An Example: A Second Attempt

min(|A|, |K−A|) 2 __ k A The cost is

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 116

An Example

An Example: A Second Attempt

The cost is at most 2 __ k A The cost is min(|A|, |K−A|) 2min(|A|, |K−A|)

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 117

An Example

An Example: A Second Attempt

< 2 2 __ k A The cost is min(|A|, |K−A|) 2min(|A|, |K−A|) The cost is at most Quality

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 118

An Example

Another Example

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 119

An Example

Another Example

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-120
SLIDE 120

An Example

Another Example

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-121
SLIDE 121

An Example

Another Example

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 122

Non-constructive Proof

Proof Outline

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-123
SLIDE 123

Non-constructive Proof

Proof Outline

1 Define a Zero-Sum Game Ankur Moitra (MIT) Sparsification September 14, 2010

slide-124
SLIDE 124

Non-constructive Proof

The Extension-Cut Game

P1 P2

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 125

Non-constructive Proof

The Extension-Cut Game

f P2 P1 A

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 126

Non-constructive Proof

The Extension-Cut Game

N(f,A)

P2 P1 A f

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 127

Non-constructive Proof

The Extension-Cut Game

N(f,A)

P2 P1 A f

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-128
SLIDE 128

Non-constructive Proof

The Extension-Cut Game

A={a}

A f

N(f,A) K−A

P2 P1

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-129
SLIDE 129

Non-constructive Proof

The Extension-Cut Game

A={a}

A f

N(f,A) K−A

P2 P1

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-130
SLIDE 130

Non-constructive Proof

The Extension-Cut Game

A={a}

P2 P1 A f

N(f,A)

K

h (a) = 5 K−A

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-131
SLIDE 131

Non-constructive Proof

The Extension-Cut Game

A={a} N(f,A)

K

h (a) = 5 K−A

P2 P1 A f

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-132
SLIDE 132

Non-constructive Proof

The Extension-Cut Game

A={a} N(f,A)

K

h (a) = 5 K−A

P2 P1 A f

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-133
SLIDE 133

Non-constructive Proof

The Extension-Cut Game

A={a} N(f,A)

K

h (a) = 5 K−A

P2 P1 A f

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-134
SLIDE 134

Non-constructive Proof

The Extension-Cut Game

A={a} N(f,A)

K

h (a) = 5 K−A

P2 P1 A f

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-135
SLIDE 135

Non-constructive Proof

The Extension-Cut Game

A={a} h (a) = 5 h (a) = 7

f

K−A

P2 P1 A f

N(f,A)

K Ankur Moitra (MIT) Sparsification September 14, 2010

slide-136
SLIDE 136

Non-constructive Proof

The Extension-Cut Game

5 A={a} N(f,A) =7 __

P2 P1 A f

N(f,A)

K

h (a) = 5 h (a) = 7

f

K−A

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 137

Non-constructive Proof

The Extension-Cut Game

___ h (A)

K f

h (A)

P2 P1 A f

N(f,A)

K

h (a) = 5 h (a) = 7

f

K−A A={a} N(f,A) =

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 138

Non-constructive Proof

Definition Let ν denote the game value of the extension-cut game

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-139
SLIDE 139

Non-constructive Proof

Definition Let ν denote the game value of the extension-cut game So ∃ a distribution γ on 0-extensions s.t. for all A ⊂ K: Ef ←γ[N(f , A)] ≤ ν

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 140

Non-constructive Proof

Definition Let ν denote the game value of the extension-cut game So ∃ a distribution γ on 0-extensions s.t. for all A ⊂ K: Ef ←γ[N(f , A)] ≤ ν Let G ′ =

f γ(f )Gf . Then for all A ⊂ K:

h′(A) =

  • f

γ(f )hf (A) = Ef ←γ[N(f , A)]hK(A) ≤ νhK(A)

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 141

Non-constructive Proof

Proof Outline

1 Define a Zero-Sum Game Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 142

Non-constructive Proof

Proof Outline

1 Define a Zero-Sum Game 2 The Best Response is a 0-Extension Problem Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 143

Non-constructive Proof

Best Response?

Let µ = 1

2{a, b} + 1 2{a, d}

d a b c

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 144

Non-constructive Proof

Best Response?

Let µ = 1

2{a, b} + 1 2{a, d}

p = d 1 __ 2 a b c

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-145
SLIDE 145

Non-constructive Proof

Best Response?

Let µ = 1

2{a, b} + 1 2{a, d}

K

_______ 1 2h ({a,b}) a b c d 1 __ 2 p = s =

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-146
SLIDE 146

Non-constructive Proof

Best Response?

Let µ = 1

2{a, b} + 1 2{a, d}

K

_______ 1 2h ({a,b}) a b c d 1 __ 2 p = s = p = 1 __ 2

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-147
SLIDE 147

Non-constructive Proof

Best Response?

Let µ = 1

2{a, b} + 1 2{a, d}

K

_______ 1 2h ({a,b}) a b c d 1 __ 2 p = s = p = 1 __ 2 s = _______ h ({a,d})

K

2 1

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-148
SLIDE 148

Non-constructive Proof

Best Response?

Let µ = 1

2{a, b} + 1 2{a, d}

K

_______ 1 2h ({a,b}) a b c d 1 __ 2 p = s = p = 1 __ 2 s = _______ h ({a,d})

K

2 1

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-149
SLIDE 149

Non-constructive Proof

Best Response?

Let µ = 1

2{a, b} + 1 2{a, d}

K

_______ 1 2h ({a,b}) a b c d 1 __ 2 p = s = p = 1 __ 2 s = _______ h ({a,d})

K

2 1

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-150
SLIDE 150

Non-constructive Proof

Proof Outline

1 Define a Zero-Sum Game 2 The Best Response is a 0-Extension Problem Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 151

Non-constructive Proof

Proof Outline

1 Define a Zero-Sum Game 2 The Best Response is a 0-Extension Problem 3 Construct a Feasible Solution for the Linear

Programming Relaxation

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 152

Non-constructive Proof

Proof Outline

1 Define a Zero-Sum Game 2 The Best Response is a 0-Extension Problem 3 Construct a Feasible Solution for the Linear

Programming Relaxation

4 Round the solution to bound the Game Value

[Fakcharoenphol, Harrelson, Rao, Talwar 2003] [Calinescu, Karloff, Rabani 2001]

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 153

Non-constructive Proof

Proof Outline

1 Define a Zero-Sum Game 2 The Best Response is a 0-Extension Problem 3 Construct a Feasible Solution for the Linear

Programming Relaxation

4 Round the solution to bound the Game Value

[Fakcharoenphol, Harrelson, Rao, Talwar 2003] [Calinescu, Karloff, Rabani 2001]

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 154

Non-constructive Proof

Bounds on the Integrality Gap

(OPT ∗ = value of the LP) Theorem (Fakcharoenphol, Harrelson, Rao, Talwar) OPT ≤ O( log k log log k )OPT ∗

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-155
SLIDE 155

Non-constructive Proof

Bounds on the Integrality Gap

(OPT ∗ = value of the LP) Theorem (Fakcharoenphol, Harrelson, Rao, Talwar) OPT ≤ O( log k log log k )OPT ∗ Theorem (Calinescu, Karloff, Rabani) If G excludes any fixed minor, OPT ≤ O(1)OPT ∗

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-156
SLIDE 156

Non-constructive Proof

Proof Outline

1 Define a Zero-Sum Game 2 The Best Response is a 0-Extension Problem 3 Construct a Feasible Solution for the Linear

Programming Relaxation

4 Round the solution to bound the Game Value

[Fakcharoenphol, Harrelson, Rao, Talwar 2003] [Calinescu, Karloff, Rabani 2001]

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 157

Thanks!

Epilogue

Ankur Moitra (MIT) Sparsification September 14, 2010

slide-158
SLIDE 158

Thanks!

Epilogue

1 Constructive results through lifting Ankur Moitra (MIT) Sparsification September 14, 2010

slide-159
SLIDE 159

Thanks!

Epilogue

1 Constructive results through lifting 2 Extensions to multicommodity flow – implications for network coding Ankur Moitra (MIT) Sparsification September 14, 2010

slide-160
SLIDE 160

Thanks!

Epilogue

1 Constructive results through lifting 2 Extensions to multicommodity flow – implications for network coding 3 Lower bounds via examples from functional analysis Ankur Moitra (MIT) Sparsification September 14, 2010

slide-161
SLIDE 161

Thanks!

Epilogue

1 Constructive results through lifting 2 Extensions to multicommodity flow – implications for network coding 3 Lower bounds via examples from functional analysis 4 Separations using harmonic analysis of Boolean functions Ankur Moitra (MIT) Sparsification September 14, 2010

slide-162
SLIDE 162

Thanks!

Thanks!

Ankur Moitra (MIT) Sparsification September 14, 2010

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SLIDE 163

Thanks!

References

1 Moitra, ”Approximation algorithms with guarantees independent of

the graph size”, FOCS 2009

2 Leighton, Moitra, ”Extensions and limits to vertex sparsification”,

STOC 2010

3 Englert, Gupta, Krauthgamer, R¨

acke, Talgam-Cohen, Talwar, ”Vertex sparsifiers: new results from old techniques”, APPROX 2010

4 Makarychev, Makarychev, ”Metric extension operators, vertex

sparsifiers and lipschitz extendability”, FOCS 2010

5 Charikar, Leighton, Li, Moitra, ”Vertex sparsifiers and abstract

rounding algorithms”, FOCS 2010

Ankur Moitra (MIT) Sparsification September 14, 2010