All-or-Nothing Multicommodity Flow

Chandra Chekuri Sanjeev Khanna Bruce Shepherd

Bell Labs U. Penn Bell Labs

Routing connections in networks

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NY – SF 10 Gb/secNY – SF 20SE – DE 5SF – DE 6

Core Optical Network

Multicommodity Routing Problem

Network – graph with edge capacities Requests: k pairs, (si, ti) with demand di

Objective: find a feasible routing for all pairs

Optimization: maximize number of pairs routed

All-or-Nothing Flow Problems

Pair is routed only if all of di satisfied

Single path for routing: unsplittable flow(connection oriented networks)

Fractional flow paths: all-or-nothing flow(packet routing networks)

Integer flow paths: all-or-nothing integer flow (wavelength paths)

Complexity of AN-Flow

di = 1 for all i

Single path: edge disjoint paths problem (EDP)

classical problem, NP-hard only polynomial approx ratios

AN-MCF: APX-hard on trees approximation ?

Approximating EDP/AN-MCF

O(min(n2/3,m1/2)) approx in dir/undir graphs (EDP/UFP) [Kleinberg 95, Srinivasan 97, Kolliopoulos-Stein 98, C-Khanna 03, Varadarajan-Venkataraman 04]

EDP is (n1/2 - )-hard to approx in directed graphs [Guruswami-Khanna-Rajaraman-Shepherd-Yannakakis 99]

LP integrality gap for EDP is (n1/2) [GVY 93]

AN-MCF: APX-hard on trees [Garg-Vazirani-Yannakakis 93]

Results

In undirected graphs AN-MCF has an O(log3 n log log n) approximation

Polynomial factor to poly-logarithmic factor

Approx via LP, integrality gap not large

For planar graphs O(log2 n log log n) approxSame ratios for arbitrary demands: dmax · umin

Online algorithm with same ratio

LP Relaxation

xi : amount of flow routed for pair (si, ti)

max i xi

s.txi flow is routed for (si,ti) 1 · i · k

0 · xi · 1 1 · i · k

A Simple Fact

Given AN-MCF instance: all di = 1

Can find (OPT) pairs such that each pair routes 1/log n flow each

How? rand rounding of LP and scaling down

Problem: we need pairs that send 1 unit each

Nice Flow Paths

Suppose all flow paths use a single vertex v

v

s1

s2

s3

s4

t1

t2

t3

t4

Routing via Clustering

v

cluster has log n terminals cluster induces a connected component clusters are edge disjoint

Clustering

Finding connected edge-disjoint clusters?

G is connected: use a spanning tree for a rough grouping of terminals

New copy of G for clustering: congestion 21 for clustering, 1 for routing

Congestion 1 using complicated clustering

How to find nice flow paths?

Algorithmic tool:

Racke’s hierarchical graph decomposition for oblivious routing [Räcke02]

Räcke’s Graph Decomposition

Represent G as a capacitated tree T

leaves of T are vertices of G

internal node v: G(v) is induced graph on leaves of T(v)

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Räcke’s Result

T is a proxy for G For all Dc*(D,G) · c(D,T) · (G) c*(D,G)

Routing in T is unique

(G) = O(log3 n) [Räcke 02](G) = O(log2 n log log n) [Harrelson-Hildrum-Rao

03]

Routing details

With each v there is distribution v on G(v) s.t

i 2 G(v) v(i) = 1

s distributes 1 unit of flow to G(v) according to v

t distributes 1 unit of flow to G(v) according to v

v

s t

Back to Nice Flow Paths

v

s1

s2

s3

s4

t1

t2

t3t4

G(v), v

s1

s2

s3

s4

t1

t2

t3t4

X(v): pairs with v as their least common ancestor (lca)

Routing in T

Routing in G

Algorithm

Find set of pairs X that can be routed in T (use tree algorithm [GVY93,CMS03])

Each pair (si,ti) in X has a level L(i) Choose level L* at which most pairs turn Route pairs independently in subgraphs

at L*

L* v

Algorithm cont’d

v at L* , X(v) pairs in X that turn at v Can route 1/(G) flow for each pair in

X(v) using nice flow paths Use clustering to route X(v)/(G) pairs

Approx ratio is (G) depth(T) = O(log3 n log log n)

Open Problems

Improve approximation ratio What is integrality gap of LP ?

No super-constant gap known

Extend ideas to EDP Recent result: Poly-log approximation for

EDP/UFP in planar graphs with congestion 3