Approximation Algorithms for Graph Routing Problems - PowerPoint Presentation

Slide1

Approximation Algorithms for Graph Routing Problems

Julia Chuzhoy

Toyota Technological Institute at ChicagoSlide2

Routing Problems

Input

: Graph G, source-sink pairs (s

1,t1),…,(sk,tk).Goal: Route as many pairs as possible; minimize edge congestion.Slide3

Routing Problems

Input

: Graph G, source-sink pairs (s

1,t1),…,(sk,tk).Goal: Route

as many pairs as possible; minimize edge congestion.Slide4

Routing Problems

Input

: Graph G, source-sink pairs (s

1,t1),…,(sk,tk).Goal: Route as many pairs as possible; minimize

edge congestion.

Edge congestion: 2Slide5

Routing Problems

Input

: Graph G, source-sink pairs (s

1,t1),…,(sk,tk).Goal: Route as many pairs as possible; minimize edge congestion.

n

– number of graph vertices

m

– number of edges

k

– number of demand pairs

terminals

– vertices participating in the demand pairsSlide6

Routing Problems

Input

: Graph G, source-sink pairs (s

1,t1),…,(sk,tk).Goal: Route as many pairs as possible; minimize edge congestion.

3 pairs with congestion 2Slide7

Routing Problems

Input

: Graph G, source-sink pairs (s

1,t1),…,(sk,tk).Goal: Route as many pairs as possible; minimize edge congestion.

3

pairs with congestion 2

2

pairs with congestion 1Slide8

Congestion MinimizationSlide9

Congestion Minimization

Route

all

demand pairsMinimize maximum edge congestion

solution value: 2Slide10

LP-relaxation

Rounding Algorithm [

Raghavan

, Thompson ‘87] Every pair (si,ti

) chooses a path P w. probability f(P)Max edge congestion O(c

log n/log log n) w.h.p.Slide11

Congestion Minimization

-

approximation

[Raghavan, Thompson ‘87]Directed graphs: -hard to approximate [Andrews, Zhang ‘06], [C, Guruswami,

Khanna, Talwar ‘07]

Undirected graphs: -hard to approximate [Andrews, Zhang ‘07]Slide12

Open Problem 1

Can we close the gap for undirected graphs?

What is the integrality gap of the LP?

What if we only need to connect a constant fraction of the demand pairs?Slide13

Edge-Disjoint Paths Problem

No congestion allowed

Route maximum number of the demand pairs

Solution value: 2Slide14

Edge Disjoint Paths (EDP)

For directed graphs NP-hard even with only two demand pairs

[Fortune,

Hopcroft, Wyllie '80]When k is constant, can be solved efficiently in undirected graphs [Robertson, Seymour ‘90]NP-hard when k is part of input [Karp ’

72]Slide15

LP-relaxationSlide16

Rounding Algorithm

[

Kolliopoulos

, Stein ‘98]Find shortest path P with non-zero flow, connecting any demand pair.Add P to the solution and delete all flow that uses edges of P.

Analysis

If the length of P is less than – at most flow is deleted.

If the length of P is more than – at most flow remains.

-approximationSlide17

Can We Do Better?

Directed graphs

: EDP is -hard

to approximate for any [Guruswami, Khanna, Rajaraman

, Shepherd, Yannakakis ‘99]Undirected graphs

: -approximation algorithm [Chekuri,

Khanna, Shepherd ’06].

-hardness of approximation for any[Andrews, Zhang ‘05], [Andrews, C, Guruswami

, Khanna, Talwar, Zhang

’10]LP integrality gap

: [Garg, Vazirani

, Yannakakis ‘93]Slide18

Integrality Gap Example[

Garg

,

Vazirani, Yannakakis ‘93]

s

1

s

2

s

k

…

t

1

t

2

t

k

…

fractional: k/2

integral: 1

k=Slide19

A Brick-Wall Graph

s

1

s

2

s

k

…

t

k

t

1

t

2

…Slide20

Open Problem 2

Close the gap for undirected EDP

planar graphs?

constant-factor approximation with congestion 2 [Chekuri, Khanna, Shepherd ‘04], [Chekuri,

Khanna, Shepherd ‘06], [Seguin-Charbonneau, Shepherd ‘11]O(log n)-approximation for Eulerian

or 4-edge connected planar graphs [Kawarabayashi

, Kobayashi ’10]constant approximation for grid-like graphs [Kleinberg,

Tardos ’95]Slide21

Open Problem 2

Close the gap for undirected EDP

planar graphs?

better algorithms for brick-wall graphs?Slide22

Positive Results

Moderately connected graphs:

If

global min-cut is Ω(log5n), there is a polylog-approximation [Rao, Zhou ‘10]

Expander graphsIn a strong enough constant-degree expander, any demand set on vertices can be routed on edge-disjoint paths [Frieze ‘00]

… Slide23

Edge Disjoint Paths (EDP)

Route maximum number of pairs on edge-disjoint paths

-approximation

matching integrality gap -hardness

Congestion MinimizationRoute all pairs; minimize congestion

-approximation -hardness

EDP with Congestion (

EDPwC

)

A factor- approximation algorithm with congestion c routes

.

demand pairs with congestion at most c.

optimum number of pairs with no congestion allowedSlide24

EDPwC

Congestion

O(log n/log log n):

constant approximation [Raghavan, Thompson ’87] -approximation with congestion c [

Azar, Regev ’

01], [Baveja, Srinivasan

’00], [Kolliopoulos, Stein ‘04]

Directed graphs: -hardness for any congestion c [C, Guruswami

, Khanna, Talwar

’06]Slide25

EDPwC

Congestion

O(log n/log log n):

constant approximation [Raghavan, Thompson ’87] -approximation with congestion c [

Azar, Regev ’

01], [Baveja, Srinivasan

’00], [Kolliopoulos, Stein ‘04]polylog

(n)-approximation with congestion poly(log log n) [Andrews ‘10]Congestion 2: -approximation [Kawarabayashi

, Kobayashi ’11]

polylog(k)-approximation with congestion 14 [C, ‘11]polylog(k)-approximation with congestion 2

[C, Li, ‘12]Slide26

Integrality Gaps for EDPwC

s

1

s

2

s

k

…

t

1

t

2

t

k

…

Congestion 1: integrality gap

[

Garg

,

Vazirani

,

Yannakakis

‘93]

Congestion c: integrality gap

[Andrews, C,

Guruswami

,

Khanna

,

Talwar

, Zhang ‘10]

-

hard

to

approximate with congestion c

for any Slide27

Edge Disjoint Paths (EDP)

Route maximum number of pairs on edge-disjoint paths

-approximation

matching integrality gap -hardness

Congestion MinimizationRoute all pairs; minimize congestion

-approximation -hardness

EDP with Congestion (

EDPwC

)

polylog

(k)-approximation with congestion 2

-hardness with congestion cSlide28

Another View: Reducing Congestion

Suppose we have a “bad” solution, where X pairs are routed with congestion C.

Then we can route X/(C

polylog k) pairs with congestion 2! But if we want congestion 1, may only be able to route pairs, even if C=2.Slide29

A Polylogarithmic

Approximation with Constant CongestionSlide30

Well-

Linkedness

[

Robertson,Seymour], [

Chekuri, Khanna, Shepherd], [Raecke

]

Graph G is well-linked for the set T of terminals,

iff

for any partition (A,B) of V(G), Slide31

Well-

Linkedness

[

Robertson,Seymour], [

Chekuri, Khanna, Shepherd], [Raecke

]

Graph G is well-linked for the set T of terminals,

iff

for any partition (A,B) of V(G), Slide32

Pre-Processing

Input

: Graph G, source-sink pairs (s

1,t1),…,(sk,tk).Theorem [Chekuri

, Khanna Shepherd ‘04]Can efficiently partition G into disjoint

subgraphs G1,…, Gr

, such that:For each induced sub-problem Gi

the terminals are well-linkedTotal fractional solution value for all induced sub-problems is

“Enough” to solve the problem on well-linked instances.Slide33

Embed an expander on a subset of terminals into G.

e

xpander

vertices terminalsexpander edges paths in GRoute a subset of the demand pairs in the expander

High-

Level Plan [CKS ‘04]

Embedding

congestion

: max load on any edge of G

CrossbarSlide34

Goal

Embed an expander over a subset of terminals into G.

Include

polylog

(k)-fraction of the terminals

congestion 2Slide35

Cut-Matching Game [

Khandekar

,

Rao

,

Vazirani

’

06]

Cut Player

: wants to build an expander

Matching Player

: wants to delay its construction

There is a strategy for cut player,

s.t.

a

fter O(log

2

n) iterations, we get an expander!Slide36

Embedding Expander into GraphSlide37

Embedding Expander into Graph

After O(log

2

k

) iterations, we get an expander embedded into G.

Problem

: congestion

Ω

(log

2

k

)Slide38

Solution?

Idea

[Rao Zhou ‘06]:Split G into graphs G1,…,Gh using algorithm of

[Karger ’93]

V(Gi)=V(G) for all iEvery edge of G belongs to at most one G

iEach Gi well-linked for the terminalsh=O(log

2k)Run the cut-matching game. Use Gi to route flow in iteration i.

Problem: Can only do it if min-cut in G is Ω(log5n)

[Andrews ‘10] adapted this to general graphs, with congestion poly(log log n) Slide39

Getting a Constant CongestionSlide40

Embedding an Expander into G

Expander vertex

connected component in G containing the terminal

Expander edge

path connecting some pair of vertices in the two components

An edge of G belongs to at most 2 of the components/paths.Slide41

Embedding an Expander into G

Routing on

vertex-disjoint

paths in X gives a congestion-2 routing in G!Slide42

Expander vertex

connected component in G containing the terminal

Expander edge

path connecting some pair of vertices in the two components

An edge of G belongs to at most 2 of the components/paths.

Embedding an Expander into GSlide43

Embedding an Expander into G

Families of Good Vertex SetsSlide44

Good Vertex Subset

S is a good vertex subset

iff

:

S contains no terminals

k/

polylog

k red edges

S is well-linked for the red edgesSlide45

A Good Family of Vertex Subsets

O(log

2

k) disjoint good vertex subsetsk/polylog k

treeseach edge of G participates in at most 2 treesEach tree Ti spans a distinct terminal

ti and a distinct red edge adjacent to Sj for each j.Slide46

Embedding an ExpanderSlide47

Embedding an Expander

Expander vertex

the tree spanning the terminal

Expander edges:

via the cut-matching game of [KRV]Slide48

Embedding an ExpanderSlide49

Embedding an Expander

After O(log

2

k) iterations, we obtain an expander embedded into G with congestion 2.

…Slide50

Algorithm for EDPwC

Find a good family of vertex subsets

Embed an expander into G

Find vertex-disjoint routing in the expander

Transform into routing in GSlide51

Open Problem 3

A cleaner algorithm for

polylog

(k) approximation with congestion 2? Better power of polylog?What if we need to route almost all demand pairs (a constant fraction)?Slide52

Routing on Vertex-Disjoint Paths

Generalizes EDP

Connections

to Graph Minor theoryNo congestion: -approximation [Kolliopoulos, Stein ‘98]

-lower bound on LP integrality gap [Garg, Vazirani

, Yannakakis ‘97]Routing with congestion:O(poly log k)-approximation with constant congestion

[Chekuri, Ene ‘12]Slide53

Better algorithms/lower bounds for Vertex-Disjoint Paths?

Open Problem 4Slide54

Better algorithms/lower bounds for Vertex-Disjoint Paths?

Vertex-disjoint paths in grid graphs

Open Problem 4Slide55

s

1

t

1

s

2

t

2

s

3

t

3

Better algorithms/lower bounds for Vertex-Disjoint Paths?

Vertex-disjoint paths in grid graphs

Open Problem 4Slide56

Thank you!