Another story onMulti-commodity Flowsand its “dual” Network

Monitoring

Rohit KhandekarIBM Watson

Joint work withBaruch Awerbuch

JHU

Outline

• Crash course:– Set cover problem and the greedy algorithm– Framework for distributed covering

problems

• The maximum multi-commodity problem and its dual passive commodity monitoring problem

• Fast converging distributed approximation schemes

The Set Cover Problem

Given• a set of elements U

• subsets S1, S2, …, Sk µ U with costs c1, c2, …, ck ¸ 0

Find• Minimum cost collection of subsets whose

union is entire U.min

Pi ci xiP

i :e2S ixi ¸ 1 8e2 Uxi 2 f0;1g 8i

The Greedy Algorithm

1. xi à 0 for all sets Si

2. re à 1 for all e2 U

3. While 9e2 U withP

i :e2S ixi < 1 do:

(a) Find a set Si that minimizesciP

e2S ire

(b) xi à 1

(c) re à 0 for all e2 Si

Gives O(log n) approximation where n = |U|.

(re = 1 if e is not yet covered)

The Fractional Set Cover Problem

minP

i ci xiPi :e2S i

xi ¸ 1 8e2 Uxi 2 f0;1g 8ixi ¸ 0

The LP relaxation of the set cover IP.

The Fractional Greedy Algorithm

1. xi à 0 for all sets Si

2. re à 1 for all e2 U

3. While 9e2 U withP

i :e2S ixi < 1 do:

(a) Find a set Si that minimizesciP

e2S ire

(b) xi à 1

(c) re à 0 for all e2 Si

Gives O(log n) ( 1 + ² ) approximation.

xi à xi + ²2

re à re ¢(1¡ ²)

Drawback: #iterations = n/²2

The Fractional Greedy Algorithm

1. xi à 0 for all sets Si

2. re à 1 for all e2 U

3. While 9e2 U withP

i :e2S ixi < 1 do:

(a) Find a set Si that minimizesciP

e2S ire

(b) xi à xi + ²2

(c) re à re ¢(1¡ ²) for all e2 Si

all

1. xi à 0 for all sets Si

2. re à 1 for all e2 U

3. While 9e2 U withP

i :e2S ixi < 1 do:

(a) Find all Si that (approx.) minimizeciP

e2S ire

(b) For all such i: do xi à xi ¢(1+ ²2) + ±

(c) Decrease re appropriately for all e

The Fractional Distributed Algorithm

# iterations =

log2(nC )²4 ¢log 1

±

Luby-Nissan (93),Garg-Konemann (98),Young (01)

Also computes a near-optimum dual solution

Concurrent Multi-commodity Flow

ce = capacity

Maximum Throughput

Concurrent Multi-commodity Flow

Send maximum total flow between the pairs subject to the edge-capacity constraints.

Maximum Throughput

Concurrent Multi-commodity Flow

Send maximum total flow between the pairs subject to the edge-capacity constraints.

Primal (packing) Dual (covering)

maxP

p f p minP

e cexe

Pp:e2p f p · ce 8e

Pe2p xe ¸ 1 8p

f p ¸ 0 8p xe ¸ 0 8e

Maximum Throughput

Distributed Computation Model

The ROUTERS model:• “Intelligence” is embodied in the network

routers• Computations takes place by exchanging

messages between neighboring routers

Complexity measures:• Approximation ratio ((1+²) approximation)• Message congestion (# messages/router/round)• Space complexity (space needed/router)• Convergence time (# rounds to converge)• Computational complexity (total work)

Multicommodity Problem & Its Dual

Primal (packing) Dual (covering)

maxP

p f p minP

e cexe

Pp:e2p f p · ce 8e

Pe2p xe ¸ 1 8p

f p ¸ 0 8p xe ¸ 0 8e

dual = set coveredges = sets

paths = elements

Dual: Probe edges e with frequency xe so that each path gets probed to an extent 1 while minimizing the total cost of probing e ce xePassive commodity monitoring

Main Result

There is an algorithm for maximum multicommodity flows and passive commodity monitoring with the following properties

• approximation

• convergence

• space and messages/router

• computational overhead

(1+ ²)

O³

log3 jP j²4

´= O

³L3 ¢logO (1) n

²4

´

~O(k ¢L) ~O(k ¢L3)

~O(m¢k ¢L3)

L = maximum hop-length of

a flowpath

Comparison with Previous Work

Reference Rounds Messages Space Computation

GK, F, Y m+ k m+ k m+ k m¢(m+ k)

LN,Y L nL nL nL

AKR, AK m¢L k ¢L k m3 ¢k ¢L

AL m¢L m¢k ¢L m¢L m2 ¢L

this work L3 k ¢L3 k ¢L m¢k ¢L3

m = number of edges

n = number of vertices

k = number of commodities

L = maximum hop-length of a °owpath

The Algorithm

• Set cover with edges as sets and paths as elements

• Associate with each path p, a residual requirement

(profit of path p)

(® is a constant)

rp = exph¡ ®¢

Pe2p xe

i

Primal (packing) Dual (covering)

maxP

p f p minP

e cexe

Pp:e2p f p · ce 8e

Pe2p xe ¸ 1 8p

f p ¸ 0 8p xe ¸ 0 8e

The Algorithm

• Repeat:

• For all edges that (approximately) minimize the cost-to-profit ratio:

increase

• Increase the flow on all paths through such edges

cePp:e2 p

rp

xe à xe(1+ ²2) + ±

How to compute aaaaaaaa

X

p

rp =X

p

Y

e2p

exp[¡ ®¢xe]

Pp:e2p rp

minp

X

e2p

leA shortest path algorithm (Dijkstra) computes:

Compute

A similar (dynamic programming) algorithm computes:

X

p

Y

e2p

le

Computing shortest paths on a “semi-ring”

(<;P

;Q

)

How to compute aaaaaaaa P

p:e2p rp

P= l1 ¢

P1 +l2 ¢

P2 +l3 ¢

P3 +l4 ¢

P4

l1

l2

l3

l4

1

2

3

4

Conclusions

• First multi-commodity algorithm– Via dual multi-cut problem– Breaks the (m) convergence barrier– Convergence polynomial in path-length L

• Question: Can we get O(L) convergence?

Thank You