Poly-Logarithmic Approximation for

EDP with Congestion 2

Julia ChuzhoyToyota Technological Institute at Chicago

Shi LiDepartment of Computer Science

Princeton Unveristy

Edge-Disjoint Path(EDP)

• Input : Graph G, k pairs (s1,t1),…, (sk, tk) of terminals

• Goal : Route as many pairs as possible using edge-disjoint paths

can route 2 pairs

s1

t1s2

t2s3

t3

s4

t4

n #verticesk #pairs terminals non-terminals

Congestion Minimization

• Input : Graph G, k pairs (s1,t1),…, (sk, tk) of terminals

• Goal : route all pairs so as to minimize the congestion

congestion = 3

s1

t1s2

t2s3

t3

s4

t4

n #verticesk #pairs terminals non-terminals

Known ResultsEdge Disjoint Paths

Congestion Minimization

Goalroute maximum # of

pairs with no congestion

route all pairs; minimize congestion

Approx.

Hardness

-integrality gap for natural LP relaxation for EDP [GVY93]

[CKS06] [RT87]

[AZ07][ACG+10]

LP Relaxation and Gap Instance

s1

s2

sk

t1 t2 tk

fractional : k/2 integral : 1k =

Edge Disjoint Path with Congestion

(EDPwC)• OPT = optimum solution for EDP

• can route at most pairs without congestion

(w.r.t the LP relaxation)

• How many pairs can we route with congestion c?

• a solution is an α-approximation for EDP with

congestion c if it routes OPT/α pairs with congestion c

Congestion

Approximation Θ(.)

c2 3 14

[ACG+10]

[Chu12]

[And10]

[AR01, BS00, KS04] [KK11]

[CKS06]

[RT87]

1

This Paper

approximationhardness

Main Theorem: There is a polylog(k)-approximation algorithm for EDP with congestion 2

We can route OPT/polylog(k) pairs with congestion 2, where OPT is the optimum number of pairs with congestion 12

Well-Linkedness• Given a set T of degree-1 terminals in graph G, • G is well-linked for T, iff for any cut (A, B) of G,

• a cluster S of G, and a subset Γ of out edges of S, define “S is well-linked for Γ ” similarly

SG

A B

Main Lemma : Given a well-linked instance, we can route k/polylog(k) pairs with congestion 2

Lemma([CKS06]) W.L.O.G, instance is well-linked:• G has degree 4• Each terminal has degree 1• Each terminal is in exactly 1 pair• G is well-linked for the set of 2k terminals

Well-Linked Instances

“cross-bar” of congestion 14

routing of k/polylog(k) pairs with congestion 14

High-Level Idea of [Chu12]Our Improvement

2

2

Cross-Bar• γ =polylog(k) disjoint clusters S1,

S2, …, Sγ

• k*=k/polylog(k) trees T1, T2,…, Tk*,

each Tj contains a terminal tj, a

edge eij in out(Si) for each i[γ]

• for each i[γ], Si is well-linked for

the set {ei1, ei2, …, eik*}

• the k* terminals {t1,t2,…,tk*} come

from k*/2 pairs of the input

S1

S2

Sγ

Si

congestion of cross bar is congestion caused by

eij

tj

Cross-Bar Construction

in[Chu12]1. select a family of clusters

with good properties

2. build trees

3. connect more terminals so that they form pairs

S1

S2

Sγ

SiProblems:

a) trees cause congestion 2 and go through clusters

b) terminals do not form pairs

our improvement: to avoid accumulating congestion, merge the 3 steps

Our AlgorithmClustering Algorithm

Cross-Bar Constructor

many paths with congestion 2

cross-bar of congestion 2

failure + feedback

disjoint “large” clusters S = (S1, S2, …, Sγ )

1. select a family of clusters with good properties

2. build trees

3. connect more terminals to trees so they form pairs

Clustering AlgorithmClustering Algorithm

Cross-Bar Constructor

many paths with congestion 2

cross-bar of congestion 2

failure + feedback

disjoint “large” clusters S = (S1, S2, …, Sγ )

Clustering Algorithm• Maintaining a clustering C (initially, singletons) of G\T into

small clusters• Each iteration

1. Randomly partition the clusters into γ parts

2. Uncontract the clusters

3. Decompose each part into well-linked clusters

CC

A cluster S is small if out(S)< k1 = k/polylog(k); otherwise, it is large

Clustering Algorithm• Maintaining a clustering C (initially, singletons) of G\T into

small clusters• Each iteration

4. If some part does not contain any large cluster

Update C by replacing that part, and start a new iteration

5. Otherwise, return γ large clusters, 1 from each part

C

if the cross-bar constructor fails, with the feedback,the clustering algorithm will make progress

Well-Linked Decomposition

• Is S well-linked for out(S)?o Yes : doneo No : find and remove the sparse

cut and recursively apply the procedure on the 2 sub-clusters

Cross-bar ConstructorClustering Algorithm

Cross-Bar Constructor

many paths with congestion 2

cross-bar of congestion 2

failure + feedback

disjoint “large” clusters S = (S1, S2, …, Sγ )

• find a degree-3 tree of clusters, o edge k/polylog(k) paths

• connect the tails of paths inside clusters of degree ≥ 2

• return the trees and degree-1 and -2 clusters

techniques:bounded degree tree [Singh Lau 07]Splitting-off [Jac98]

Cross-bar Constructor :

High-Level Idea

Summary• A polylog(k)-approximation for EDP with congestion

2.• Future directions:

o Simpler algorithm?o Better approximation for EDP with congestion 1?

(interesting even for planar graphs)o Better approximation or better hardness for congestion

minimization problem?

Thank you