Approximation Algorithms for Min-Max Resource Sharing ...oplab/seminar/2004/huzhang.pdf · Overview • Min-max resource sharing problems, approximate block solver • Potential function,

Approximation Algorithms

for Min-Max Resource Sharing Problems

and Its Application

in the Multicast Congestion Problem

Hu Zhang

Joint work with Klaus Jansen

McMaster Optimization Seminar

November 15, 2004

1

Overview

• Min-max resource sharing problems, approximate block solver

• Potential function, price vector

• Main ideas of the approximation algorithms

• Analysis of the approximation algorithms

• The multicast congestion problem in communication networks

• Future work

2

Min-Max Resource Sharing (MMRS) Problems

min λ

s.t. fm(x1, . . . , xN) ≤λ, m = 1, . . . ,M ;

(x1, . . . , xN)T ∈ B,

where B ⊆ IRN is a nonempty convex compact set and

fm : B → IR+ are nonnegative continuous convex functions on B

for m ∈ {1, . . . ,M}. Let f(x)= (f1(x), . . . , fM(x))T and

λ(x)= maxm∈{1,...,M} fm(x) for x = (x1, . . . , xN)T ∈ B.

3

Duality Relation

λ∗= minx∈B

maxp∈P

pT f(x) = maxp∈P

minx∈B

pT f(x),

where P = {(p1, . . . , pM)T |∑M

m=1 pm = 1, pm ≥ 0}.

Let Λ(p)= minx∈B pT f(x).

• For any pair x ∈ B, p ∈ P it holds: Λ(p) ≤ λ∗ ≤ λ(x).

• A pair x ∈ B and p ∈ P is optimal, iff λ(x) = Λ(p).

4

Approximate Problem

Goal: compute an x ∈ B such that f(x) ≤ c(1 + ε)λ∗ ·~1.

Block problem: Λ(p) = min{pT f(x)|x ∈ B} where p ∈ P .

Block solver ABS(p,t,c): compute an x ∈ B such that

pT f(x) ≤ c(1 + t)Λ(p).

5

Main Results I

Primal and dual problem

(Pc,ε) compute an x ∈ B such that f(x) ≤ c(1 + ε)λ∗ ·~1,

(Dc,ε) compute a p ∈ P such that Λ(p) ≥ 1c(1 − ε)λ∗.

Theorem: For any ε > 0 there is an algorithm that solves (Pc,ε) and

(Dc,ε) in O(M(ε−2 + ln M + ε−3 ln c)) iterations (calls to the

block solver), provided that there is an approximate block solver

ABS(p,O(ε), c) for any p ∈ P .

6

Main Results II

Theorem: There is an algorithm that for any relative error tolerance

ε ∈ (0, 1) solves the problem (Pc,ε) in

O(M(ln M + ε−2 ln ε−1))

iterations, provided that there is an approximate block solver

ABS(p,O(ε), c) for any p ∈ P .

Theorem: If ln c = O(ε), then O(M(ln M + ε−2)) iterations are

necessary to solve both (Pc,ε) and (Dc,ε).

7

Previous Results

• for linear functions and c = 1: O(ε−2ρ ln(Mε−1)) iterations, (Plotkin,

Shmoys and Tardos 91) .

• for linear functions: O(ε−2ρ′(λ∗)−1 lnM) iterations, (Young 95) .

• for convex functions and c = 1: O(M(lnM + ε−2 ln ε−1)) iterations,

(Grigoriadis and Khachiyan 96), (Villavicencio and Grigor iadis 97) .

• for linear functions and c = 1: O(Mε−2 lnM) iterations, (Garg and

Konemann 98) .

• for linear functions: O(ε−2ρ ln(Mε−1)) iterations, (Charikar, Chekuri,

Goel, Guha and Plotkin 98) .

8

Results for Max-Min Resource Sharing Problems

• for linear functions and c = 1: O(M + ρ ln2 M + ε−2ρ ln(Mε−1))

iterations, (Plotkin, Shmoys and Tardos 91) .

• for concave functions and c = 1: O(M(lnM + ε−2 ln ε−1)) iterations,

(Grigoriadis, Khachiyan, Porkolab and Villavicencio) .

• for general case: O(M(ε−2 + lnM + ε−3 ln c)) iterations, (Jansen and

Porkolab 02) .

• for general case: O(Mε−2 ln(Mε−1)) iterations, (Jansen 02) .

• for linear functions and c = 1: O(Mε−2 ln(MC)) iterations, (Fleischer 04) .

• for linear functions and c = 1: O(Mε−2 lnM + min{N, ln lnC})

iterations, (Garg and Khandekar 04) .

• for general case: O(M(lnM + ε−2 ln ε−1)) iterations, (Jansen 04) .

9

Results for Mixed Problems

• O(Mdε−2 ln M) iterations, (Young 01) .

• O(Mε−2 ln(Mε−1)) iterations, (Jansen 04) .

• O(M(ln M + ε−2 ln ε−1)) iterations, (Jansen 04) .

10

Logarithmic Potential Function

Φt(θ, x)= ln θ −t

M

M∑

m=1

ln(θ − fm(x)),

where θ ∈ IR, x ∈ B and t > 0 is the parameter in ABS(p, t, c).

Φt is well-defined for λ(x) < θ < ∞.

The minimizer θ(x) can be determined from

tθ

M

M∑

m=1

1

θ − fm(x)= 1.

based on (Villavicencio and Grigoriadis 97) .

11

Properties of the Potential Function

Lemma:λ(x)

(1 − t/M)≤ θ(x) ≤

λ(x)

(1 − t).

Lemma:

(1 − t) ln λ(x) ≤ φt(x) ≤ (1 − t) ln λ(x) + t ln(e/t),

where φt(x)= Φt(θ(x), x) is the reduced potential function.

12

Price Vector

The price (dual) vector p(x) = (p1(x), . . . , pM(x))T is given by:

pm(x) =t

M

θ(x)

θ(x) − fm(x), m = 1, . . . ,M.

Lemma:

(1) p(x) ∈ P ,

(2) p(x)T f(x) = (1 − t)θ(x).

13

Coordination

Step 1: compute the price vector p ∈ P using current iterate x ∈ B;

Step 2: call a block solver to compute an (approximate) solution x ∈ B

using price vector p;

Step 3: move from x to (1 − τ)x + τ x with a step length τ ∈ (0, 1] for

the new iterate x.

14

Coordination

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15

Scaling Phase

• the relative error tolerance σ1 = 1 in the first phase;

• halve the relative error tolerance in each phase (σs = σs−1/2);

• the relative error tolerance σsN≤ ε in the last phase.

16

Stopping Rule 1

Let x ∈ B be the solution of ABS(p, σs/6, c). We define

ν(x, x) =p(x)T f(x) − p(x)T f(x)

p(x)T f(x) + p(x)T f(x).

Rule 1:

ν(x, x) ≤ σs/6.

17

Stopping Rule 2

For phase s we define the parameter:

ws =

1+σ1

(1+σ1/6)M, for s = 1;

1+σs

1+2σs, otherwise.

Rule 2:

λ(x) ≤ ws λ(x(s−1)).

18

Flowchart

Start

Initialization

End

LagrangianCoordination

ScalingTolerance

Output

StoppingRule 1 or 2?

AccuracyAchieved?

No!

No!

Yes!

Yes!

19

Approximation Algorithm

(1) compute initial solution x(0), s := 0, σ1 := 1;

(2) repeat {scaling phase }

(2.1) s := s + 1; x := x(s−1); finished := false;

(2.2) while not(finished) do

(2.2.1) compute θ(x) and p(x);

(2.2.2) x := ABS(p(x), σs/6, c);

(2.2.3) if either stopping rule 1 or 2 is satisfied

then finished := true; x(s) := x;

else x = (1 − τ)x + τ x;

(2.3) σs+1 = σs/2 until σs ≤ ε;

(3) return(x(s)).

20

Initial Solution

Let x(0) ∈ B be the solution of ABS(~1/M, 1/6, c).

Lemma:

λ(x(0)) ≤ (7/6)cMλ∗.

Proof:

fm(x(0)) ≤∑M

m=1 fm(x(0)) = M(~1/M)T f(x(0))

≤ M(7/6)cΛ(~1/M) ≤ (7/6)cMλ∗.

This implies that λ(x(0)) = maxm fm(x(0)) ≤ (7/6)cMλ∗.

21

Analysis I

Lemma: If the algorithm stops, then the computed vector x(s) solves

the problem (Pc,ε).

Proof: consider two different cases:

Case 1: Stopping rule 1 is satisfied. Since ν(x(s), x(s)) ≤ ts = σs/6, we have

(1 − ts)p(x(s))T f(x(s)) ≤ (1 + ts)p(x(s))T f(x(s)). This implies

θ(x(s)) = p(x(s))T f(x(s))1−ts

≤ (1+ts)p(x(s))T f(x(s))(1−ts)2

≤ c (1+ts)2

(1−ts)2 Λ(p) ≤ c(1 + σs)λ∗.

Therefore,

fm(x(s)) ≤ λ(x(s)) ≤ θ(x(s)) ≤ c(1 + σs)λ∗.

22

Analysis I

Case 2: Stopping rule 2 is satisfied.

Case 2.1: s = 1. Then, λ(x(1)) ≤ (1+σ1)(1+σ1/6)M λ(x(0)).

Since λ(x(0)) ≤ (1 + σ1/6)cMλ∗, we get λ(x(1)) ≤ (1 + σ1)cλ∗.

Case 2.2: s > 1. By induction λ(x(s−1)) ≤ (1 + σs−1)cλ∗. Using

σs = σs−1/2,

λ(x(s)) ≤(1 + σs)

(1 + 2σs)λ(x(s−1)) ≤ (1 + σs)cλ

∗.

23

Analysis II

Lemma: For any two consecutive iterates x, x′ ∈ B within scaling

phase s, φts(x′) ≤ φts(x) − tsν

2/(4M), where ts = σs/6.

Theorem: For any ε ∈ (0, 1), our algorithm needs at most

O(M(ln M + ε−2 ln ε−1)) iterations.

Proofidea: Let Ns be the number of iterations in phase s, let x be the

initial solution of phase s.

24

Proofidea

(a) φts(x) − φts(x) ≥ (Ns − 1)t3s/(4M) where x is the solution

after Ns − 1 iterations;

(b) φts(x) − φts(x) ≤ (1 − ts) ln λ(x)λ(x)

+ ts ln ets

using the lower

and upper bounds for the reduced potential function;

(c) if s = 1 thenλ(x)λ(x)

≤ 2M1+σ1

. This gives N1 = O(M ln M);

(d) if s > 1 thenλ(x)λ(x)

≤ 1 + σs

1+σs. This gives

Ns = O(Mσ−2s ln σ−1

s ).

25

Special Case of Small c

For ln c = O(ε), strategy: only Stopping Rule 1.

Lemma: For any two arbitrary iterates x, x′ ∈ B within scaling phase

s, φts(x) − φts(x′) ≤ (1 − ts) ln p(x)T f(x)

Λ(p(x)).

Theorem: For any ε ∈ (0, 1), our algorithm needs at most

O(M(ln M + ε−2)) iterations.

26

Applications

• Scheduling unrelated machines (Plotkin, Shmoys & Tardos 91), (Jansen 01) .

• Job shop scheduling (Plotkin, Shmoys & Tardos 91) .

• Network embeddings (Plotkin, Shmoys & Tardos 91) .

• Held-Karp bound for TSP (Plotkin, Shmoys & Tardos 91) .

• Multicommodity flows (Grigoriadis & Khachiyan 96), (Garg & K onemann 98) .

• Bin covering (Jansen & Solis-Oba 02) .

• Spreading metrics (Even, Naor, Rao & Schieber 99) .

• Approximating metric spaces/edge weighted graphs (Charikar, Chekuri, Goel,

Guha & Plotkin 98), (Fakcharoenphol, Rao & Talwar 03), (Chle bıkov a, Ye &

Z. 04).

• Multicast congestion in communication networks (Jansen & Z. 02) .

• Range assignment in ad-hoc networks (Ye & Z. 04), (Chlebıkov a, Ye & Z. 04).

27

Multicast Edge Congestion

Given:

• a communication network G = (V,E),

• k multicast requests S1, . . . , Sk ⊆ V .

Solution: set of trees T1, . . . , Tk where Ti spans the vertex set Si

for i = 1, . . . , k.

Goal: minimize the maximum edge congestion (number of trees that

use an edge).

28

Example - Multicast Congestion

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Example - Multicast Congestion

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30

Related Graph Problems

The multicast congestion problem is related to:

• integral paths with minimum edge congestion (where |Si| = 2 for

each 1 ≤ i ≤ k) that generalize itself the edge disjoint path

problem),

• minimum Steiner trees in graphs.

31

Integer Linear Programming

min λ

s.t.∑k

i=1

∑

T∈Ti,e∈T xi(T ) ≤λ, e ∈ E;∑

T∈Tixi(T ) = 1, i = 1, . . . , k;

xi(T ) ∈ {0, 1}, i = 1, . . . , k, T ∈ Ti,

where Ti is the set of all trees spanning Si for i = 1, . . . ,m and

xi(T ) is the indicator variable.

Main Approach:

• solving LP-relaxation;

• rounding.

32

Packing Formulation

min λ

s.t.∑k

i=1

∑

T∈Ti,e∈T xi(T ) ≤λ, e ∈ E,

(xi(T )) ∈ B,

where

B = {(xi(T ))|∑

T∈Ti

xi(T ) = 1, xi(T ) ≥ 0, T ∈ Ti, i = 1, . . . , k}.

33

Block Problem

Λ(p) = minx∈B pT f(x) with p ∈ P is an instance of the minimum

Steiner tree problem.

• NP-hard , even for unweighted graphs (Karp 72) .

• c = 2, β = O(m + n lnn) (Moore (cf. Gilbert and Pollak 68), Mehlhorn

88, Floren 91) .

• c = 11/6, β = O(mn + s4) (Zelikovsky 93) .

• c = 1 + ln 3/2 ≈ 1.550, β polynomial (Robins and Zelikovsky 00) .

• there exists a constant c > 1 such that there is no polynomial-time

approximation algorithm with c < c (Bern and Plassmann 89) .

• c ≥ 96/95 ≈ 1.0105 (Chlebık and Chlebıkov a 02).

where β is the running time.

34

Running Times of LP-Relaxation

• O(n6α2 + n7α) (where α involves k and other logarithmic terms) (Vempala

and Vocking 99) .

• O(n7) (Carr and Vempala 00) .

• O(βnk3 ln3(mε−1)ε−9 min{lnm, ln k}) (Baltz and Srivastav 01) .

• O(βk2 ln(mε−1)ε−2 ln(kε−1)) using the approach in (Charikar, Chekuri,

Goel, Guha and Plotkin 98) .

• O(m(lnm + ε−2 ln ε−1)(βk + m ln ln(mε−1))) (Jansen and Z. 02) .

• O(k(m + β)ε−2 ln k lnm) (Baltz and Srivastav 03) .

35

Future Work

• Theoretical work:

– negative constraints;

– block structure of B;

– max-min resource sharing problems and mixed problems;

– . . .

• Applications:

– routing in VLSI design;

– routing in optical networks;

– cylindrical reconstruction;

– maximum independent set in t-perfect graphs;

– . . .

• Implementation.

36

Approximation Algorithms for Min-Max Resource Sharing ...oplab/seminar/2004/huzhang.pdf · Overview • Min-max resource sharing problems, approximate block solver • Potential function,

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