A Unified Approach to A Unified Approach to ApproximatingApproximating

Resource Allocation and Resource Allocation and SchedulingScheduling

Amotz Bar-Noy.……...AT&T and Tel Aviv University

Reuven Bar-Yehuda….Technion IIT

Ari Freund……………Technion IIT

Seffi Naor…………….Bell Labs and Technion IIT

Baruch Schieber…...…IBM T.J. Watson

Slides and paper at:

http://www.cs.technion.ac.il/~reuven

Summery of Results:Summery of Results:DiscreteDiscrete

Single Machine Scheduling

Bar-Noy, Guha, Naor and Schieber STOC 99: 1/2 Non Combinatorial*

  Berman, DasGupta, STOC 00: 1/2

  This Talk, STOC 00(Independent)      1/2

Bandwidth AllocationAlbers, Arora, Khanna SODA 99: O(1) for |Activity i|=1*Uma, Phillips, Wein SODA 00:        1/4 Non combinatorial.This Talk STOC 00 (Independent)       1/3 for w 1/2 This Talk STOC 00 (Independent)       1/5 for w 1   

Parallel Unrelated Machines:

Bar-Noy, Guha, Naor and Schieber STOC 99: 1/3

  Berman, DasGupta STOC 00: 1/2

  This Talk, STOC 00(Independent)      1/2

Summery of Results:Summery of Results:ContinuousContinuous

Single Machine SchedulingBar-Noy, Guha, Naor and Schieber STOC 99: 1/3 Non Combinatorial

  Berman, DasGupta STOC 00: 1/2·(1-)

  This Talk, STOC 00: (Independent) 1/2·(1-)

Bandwidth AllocationUma, Phillips, Wein SODA 00:       1/6 Non combinatorial

This Talk, STOC 00 (Independent)       1/3 ·(1-) for w 1/2        1/5 ·(1-) for w 1

Parallel unrelated machines:Bar-Noy, Guha, Naor and Schieber STOC 99: 1/4

  Berman, DasGupta STOC 00: 1/2·(1-)

  This Talk, STOC 00: (Independent) 1/2·(1-)

Summery of Results:Summery of Results:and more…and more…

General Off-line Caching Problem  Albers, Arora, Khanna SODA 99: O(1) Cache_Size += O(Largest_Page)

   O(log(Cache_Size+Max_Page_Penalty))

This Talk, STOC 00: 4

Ring topology: Transformation of approx ratio from line to ring topology 1/ 1/(+1+)

Dynamic storage allocation (contiguous allocation): Previous results: none for throughput maximization Previous results Kierstead 91 for resource minimization: 6 This paper: 1/35 for throughput max using the result for resource min.

The Local-Ratio Technique:The Local-Ratio Technique: Basic definitions Basic definitions

Given a profit [penalty] vector p.

Maximize[Minimize] p·x

Subject to: feasibility constraints F(x)

x is r-approximation if F(x) and p·x [] r · p·x*

An algorithm is r-approximation if for any p, F

it returns an r-approximation

The Local-Ratio Theorem:The Local-Ratio Theorem:

x is an r-approximation with respect to p1

x is an r-approximation with respect to p- p1

x is an r-approximation with respect to p

Proof: (For maximization)

p1 · x r × p1*

p2 · x r × p2*

p · x r × ( p1*+ p2*)

r × ( p1 + p2 )*

Special case: Optimization is 1-approximationSpecial case: Optimization is 1-approximation

x is an optimum with respect to p1

x is an optimum with respect to p- p1

x is an optimum with respect to p

A Local-Ratio Schema for A Local-Ratio Schema for Maximization[Minimization] problems: Maximization[Minimization] problems:

Algorithm r-ApproxMax[Min]( Set, p )

If Set = Φ then return Φ ;

If I Set p(I) 0 then return r-ApproxMax( Set-{I}, p ) ;

[If I Set p(I)=0 then return {I} r-ApproxMin( Set-{I}, p ) ;]

Define “good” p1 ;

REC = r-ApproxMax[Min]( S, p- p1 ) ;

If REC is not an r-approximation w.r.t. p1 then “fix it”;

return REC;

The Local-Ratio Theorem: ApplicationsThe Local-Ratio Theorem: Applications

Applications to some optimization algorithms (r = 1):

( MST) Minimum Spanning Tree (Kruskal)

( SHORTEST-PATH) s-t Shortest Path (Dijkstra)

(LONGEST-PATH) s-t DAG Longest Path (Can be done with dynamic programming)

(INTERVAL-IS) Independents-Set in Interval Graphs Usually done with dynamic programming)

(LONG-SEQ) Longest (weighted) monotone subsequence (Can be done with dynamic programming)

( MIN_CUT) Minimum Capacity s,t Cut (e.g. Ford, Dinitz)

Applications to some 2-Approximation algorithms: (r = 2)

( VC) Minimum Vertex Cover (Bar-Yehuda and Even)

( FVS) Vertex Feedback Set (Becker and Geiger)

( GSF) Generalized Steiner Forest (Williamson, Goemans, Mihail, and Vazirani)

( Min 2SAT) Minimum Two-Satisfibility (Gusfield and Pitt)

( 2VIP) Two Variable Integer Programming (Bar-Yehuda and Rawitz)

( PVC) Partial Vertex Cover (Bar-Yehuda)

( GVC) Generalized Vertex Cover (Bar-Yehuda and Rawitz)

Applications to some other Approximations:

( SC) Minimum Set Cover (Bar-Yehuda and Even)

( PSC) Partial Set Cover (Bar-Yehuda)

( MSP) Maximum Set Packing (Arkin and Hasin)

Applications Resource Allocation and Scheduling :

….

Maximum Independent Set in Interval GraphsMaximum Independent Set in Interval Graphs

Activity9Activity8Activity7Activity6Activity5Activity4Activity3Activity2

Activity1

time

Maximize s.t. For each instance I:

For each time t:

I

IxIp )( }1,0{Ix

)()(:

1IetIsIIx

Maximum Independent Set in Interval Graphs: Maximum Independent Set in Interval Graphs: How to select How to select PP11 to get optimization? to get optimization?

Activity9Activity8Activity7Activity6Activity5Activity4Activity3Activity2

Activity1 Î time

Let Î be an interval that ends first;

1 if I in conflict with Î

For all intervals I define: p1 (I) = 0 else

For every feasible x: p1 ·x 1

Every Î-maximal is optimal.

For every Î-maximal x: p1 ·x 1

P1=1

P1=1

P1=1

P1=1

P1=0

P1=0

P1=0

P1=0

P1=0

Maximum Independent Set in Interval Graphs:Maximum Independent Set in Interval Graphs: An Optimization Algorithm An Optimization Algorithm

Activity9Activity8Activity7Activity6Activity5Activity4Activity3Activity2Activity1 Î

time

Algorithm MaxIS( S, p )If S = Φ then return Φ ;

If I S p(I) 0 then return MaxIS( S - {I}, p);Let Î S that ends first;

I S define: p1 (I) = p(Î) (I in conflict with Î) ;

IS = MaxIS( S, p- p1 ) ;If IS is Î-maximal then return IS else return IS {Î};

P1=0

P1=0

P1=0

P1=0

P1=0

P1=P(Î )

P1=P(Î )

P1=P(Î )

P1=P(Î )

Maximum Independent Set in Interval Graphs:Maximum Independent Set in Interval Graphs: Running Example Running Example

P(I1) = 5 -5

P(I4) = 9 -5 -4

P(I3) = 5 -5

P(I2) = 3 -5

P(I6) = 6 -4 -2

P(I5) = 3 -4

-5 -4 -2

Single Machine Scheduling :Single Machine Scheduling :

Activity9

Activity8

Activity7

Activity6

Activity5

Activity4

Activity3

Activity2

Activity1 ?????????????

time

Maximize s.t. For each instance I:

For each time t:

For each activity A:

I

IxIp )( }1,0{Ix

)()(:

1)(IetIsI

IxIw

1AI

Ix

Bar-Noy, Guha, Naor and Schieber STOC 99: 1/2 LP

Berman, DasGupta, STOC 00: 1/2

This Talk, STOC 00(Independent)      1/2

Single Machine Scheduling:Single Machine Scheduling: How to select How to select PP11 to get ½-approximation ? to get ½-approximation ?

Activity9Activity8Activity7Activity6Activity5Activity4Activity3Activity2

Activity1 Î time

Let Î be an interval that ends first;

1 if I in conflict with Î For all intervals I define: p1 (I) =

0 else

For every feasible x: p1 ·x 2

Every Î-maximal is 1/2-approximation

For every Î-maximal x: p1 ·x 1

P1=1P1=1P1=1P1=1

P1=1

P1=1

P1=1

P1=1

P1=1

P1=0

P1=0

P1=0

P1=0

P1=0

P1=0

P1=0

P1=0

P1=0 P1=0

P1=0

Single Machine Scheduling:Single Machine Scheduling: The ½-approximation Algorithm The ½-approximation Algorithm

Activity9Activity8Activity7Activity6Activity5Activity4Activity3Activity2Activity1 Î

time

Algorithm MaxIS( S, p )If S = Φ then return Φ ;

If I S p(I) 0 then return MaxIS( S - {I}, p);Let Î S that ends first;

I S define: p1 (I) = p(Î) (I in conflict with Î) ;

IS = MaxIS( S, p- p1 ) ;

If IS is Î-maximal then return IS else return IS {Î};

Activity9Activity8Activity7Activity6Activity5Activity4Activity3Activity2Activity1 I w(I)

s(I) e(I) time

Maximize s.t. For each instance I:

For each time t:

For each activity A:

BandwidthBandwidth Allocation Allocation

I

IxIp )(

)()(:

1)(IetIsI

IxIw

1AI

Ix

}1,0{Ix

Albers, Arora, Khanna SODA 99: O(1) |Ai|=1*

Uma, Phillips, Wein SODA 00: 1/4 LP.

This Talk 1/3 for w 1/2 and 1/5 for w 1  

BandwidthBandwidth Allocation for Allocation for ww 1/2 1/2 How to select How to select PP11 to get 1/3-approximation? to get 1/3-approximation?

Activity9

Activity8 Î

Activity7

Activity6

Activity5

Activity4

Activity3

Activity2

Activity1 I w(I)

s(I) e(I) time

1 if I in the same activity of Î

For all intervals I define: p1 (I) = 2*w(I) if I in time conflict with Î

0 else

For every feasible x: p1 ·x 3

Every Î-maximal is 1/3-approximation

For every Î-maximal x: p1 ·x 1

BandwidthBandwidth Allocation Allocation The 1/5-approximation for any The 1/5-approximation for any w w 1 1

Activity9Activity8 w > ½ Activity7 w > ½ w > ½Activity6Activity5 w > ½Activity4Activity3 w > ½ w > ½Activity2Activity1 w > ½ w > ½ w > ½

Algorithm:1. GRAY = Find 1/2-approximation for gray (w>1/2) intervals;2. COLORED = Find 1/3-approximation for colored intervals3. Return the one with the larger profitAnalysis:If GRAY* 40%OPT then GRAY 1/2(40%OPT)=20%OPT elseCOLORED* 60%OPT thus COLORED 1/3(60%OPT)=20%OPT

Continuous SchedulingContinuous Scheduling

Single Machine Scheduling (w=1)

Bar-Noy, Guha, Naor and Schieber STOC 99: 1/3 Non Combinatorial

  Berman, DasGupta STOC 00: 1/2·(1-)

  This Talk, STOC 00: (Independent) 1/2·(1-)

Bandwidth AllocationUma, Phillips, Wein SODA 00:       1/6 Non combinatorial

This Talk, STOC 00 (Independent)       1/3 ·(1-) for w 1/2        1/5 ·(1-) for w 1

s(I) e(I)

{w(I) d(I)

If currant p(I1) original p(I1) then delete I1

else Split I2=(s2,e2] to I21=(s2, s1+d1] and I22=(s1+d1,e2]

Continuous Scheduling:Continuous Scheduling: Split and Round Profit Split and Round Profit (Loose additional (1- (Loose additional (1-) factor)) factor)

d(I1)

d(I2)

d(I1)

d(I2)

I11 I12

I21 I22

d

d

d

d

d

d

d

d

d

d

Parallel Unrelated Machines: ContinousBar-Noy, Guha, Naor and Schieber STOC 99: 1/3 1/4

  Berman, DasGupta STOC 00: 1/2 1/2·(1-)

  This Talk, STOC 00(Independent)      1/2 1/2·(1-)

Parallel unrelated machines:Parallel unrelated machines:

c

d

d

dd

h

h

h

cc

c

k

A

Ai

i

ik

Parallel unrelated machines: 1/5-approximation Parallel unrelated machines: 1/5-approximation (not in the paper)(not in the paper) Each machine resource Each machine resource 11pp11((Red Red ) = ) = pp11((orange orange d ) = 1;) = 1;

pp11 ((YellowYellow d ) = 2width; ) = 2width; pp11 (All others) = 0; (All others) = 0;

c

d

d

d

d

d

d

d

h

h

h

cc

c

k

A

Ai

i

ik