Lecture 11 Overview Self-Reducibility. Overview on Greedy Algorithms

Lecture 11

Overview

Self-Reducibility

Overview on Greedy Algorithms

strategy.greedy thefrom followsproperty

exchange themeanwhile ty,reducibili-self andproperty

exchangewith together foundoften is algorithmgreedy The

Revisit Minimum Spanning Tree

Exchange Property

tree.spanning minimum a

still is )\(such that in edgean exist

must there, without treespanning minimum a and

graph ain eight smallest w with the edgean For

e e'TTe'

eT

Ge

Self-Reducibility

.' of treespanning minimum a is Then point. a

into shrinkingby ly respective , and from

obtained be and Let . of edgean is and

graph a of treespanning minimum a is Suppose

GT'

eTG

T'G'Te

GT

Max Independent Set in Matroid

Exchange Property

tree.spanning minimum a

still is )\(such that in edgean exist

must there, without set t independen minimum a and

matroid ain eight smallest w with theelement an For

e e'TTe'

eT

Ge

Self-Reducibility

.' of treespanning minimum a is Then point. a

into shrinkingby ly respective , and from

obtained be and Let . of edgean is and

graph a of treespanning minimum a is Suppose

GT'

eTG

T'G'Te

GT

etc.. Ratio, Local as

such ,algorithms of sother type ofdesign in used

also isty reducibili-self The ty.reducibili-self

using algorithms of pespopular ty threeare

Greedy Program, Dynamic Conquer,-and-Divide

Overview on Greedy Algorithms

Exchange Property

Matroid

Self-Reducibility

Local Ratio Method

Basic Idea

*)(*)(*)()()()(

*)()( *),()(

.function objectivefor solution optimalan is *then

,function objectivefor solution optimalan also and

function objectivefor solution optimalan is * If

).()()( Suppose

)(maxor )(minx

problemtion optimimizaan Consider

2121

2211

2

1

21

xcxcxcxcxcxc

xcxcxcxc

cx

c

cx

xcxcxc

xcxcxx

Proof

Basic Idea

).(min toreduced is

problem thesolutions, optimal ofset large a has )(When

.function objectivefor solution optimalan is *then

,function objectivefor solution optimalan also and

function objectivefor solution optimalan is * If

).()()( Suppose

)(maxor )(minx

problemtion optimimizaan Consider

2

1

2

1

21

xc

xc

cx

c

cx

xcxcxc

xcxc

x

xx

Minimum Spanning Tree

1

56

3 4

2

7 5

1

1 1

1 1

1

1 1

0

4 4

1

2 3

5 6

solution! optimalan

is treespanningEvery

1

44

5 6

2

Activity Selection

.),[),[ : pingnonoverlap

y.cardinalit themaximize tointervals

pingnonoverlap ofsubset a find ),,[

),...,,[),,[ intervals ofset aGiven 2211

jjii

nn

fsfs

fs

fsfs

Puzzle

solution? optimalan find way toefficient an findyou can not, If

hold? stillproperty exchange theDoes

intervals.

pingnonoverlap weight totalmaximum thefind want to weandweight

enonnegativ a has intervaleach suppose problem,selection -activityIn

17

Independent Set in Interval Graphs

Activity 9Activity 8Activity 7Activity 6Activity 5Activity 4Activity 3Activity 2Activity 1

• We must schedule jobs on a single processor with no preemption. • Each job may be scheduled in one interval only.• The problem is to select a maximum weight subset of non-conflicting

jobs.

time

18

Independent Set in Interval Graphs

I

IxIp )( }1,0{Ix

)()(:

1IftIsIIx

Activity9Activity8Activity7Activity6Activity5Activity4Activity3Activity2 Activity1

Maximize s.t. For each instance I

For each time t

time

Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

19

Maximal Solutions

• We say that a feasible schedule is I-maximal if either it contains instance I, or it does not contain I but adding I to it will render it infeasible.


time

I2I1

The schedule above is I1-maximal and also I2-maximal

20

An effective profit function

P1= P(Î)

P1=0

P1=0

P1=0

P1=0

P1=0


Let Î be an interval that ends first;

Î

P1= P(Î)

P1= P(Î)

P1= P(Î)


negative.) becan )( :(note )()()(

otherwise 0

ˆith conflect win if )ˆ()(

212

1

IpIpIpIp

IIIpIp

21

An effective profit function

P1= P(Î)

P1=0

P1=0

P1=0

P1=0

P1=0


Î

P1= P(Î)

P1= P(Î)

P1= P(Î)

For every feasible solution x: p1 ·x p(Î) For every Î-maximal solution x: p1 ·x p(Î)

Every Î-maximal is optimal.


22

Independent Set in Interval Graphs:An Optimization Algorithm

Algorithm MaxIS( S, p )1. If S = Φ then return Φ ;2. If I S p(I) 0 then return MaxIS( S - {I}, p);3. Let Î S that ends first;4. I S define: p1 (I) = p(Î) (I in conflict with Î) ;5. IS = MaxIS( S, p- p1 ) ;6. If IS is Î-maximal then return IS else return IS {Î};


23

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


Minimum Weight Arborescence

Definition

.other every to frompath a is There (b)

tree.spanning a is

then ignored, are edges theof directions theIf (a)

hold. conditions following thesuch that and edges

opposite ofpair acontain not does which ),( of

),(subgraph a is root with An

Vvr

T

EVG

FV T rcearborescen

Problem

cost. totalminimum with earborecenc

an find , node a and :cost edge

enonnegativ with ),(graph directed aGiven

VrREc

EVG

Key Point 1

. nodeat edges-in ofset the)( vvin

. cycle a contains * Otherwise, optimal. is

* then ce,arborescenan is }}{|{* If

).( from edgecheapest a choose ,each For

CF

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verv

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Key Point 2

. nodeat edges-in ofset the)( vvin

. cycle a contains * Otherwise, optimal. is

* then ce,arborescenan is }}{|{* If

).( from edgecheapest a choose ,each For

CF

FrVveF

verv

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'.' .cost w.r.tminiman isit iff

cost w.r.t minimum is cearborescenan Then,

).,('),(),('' and

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c

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Why?

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optimalan is cearborescenany that means This

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Key Point 3

once.exactly enters which ),(

cearborescencost -minimum a exists Then there

.0)(''with containingnot cycle a be Let

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A Property of MST

.)(\*)(in edges least

at contains ,*)(\)(subset any for :Claim

. ,in edgeevery fore and edge contains

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where,)(),(every for such that *)()( :

mapping onto one-to-one a is Then there ly.respective network, a of

treespanning minimum theand treespanning a be * and Let

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tree.a is that ingcontradict ,

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from mapping one-to-one a exists thereTheorem, Marriage sHall'By

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*).()()(

Therefore, .)(

hence ),(such that )( exists theredifferent,

are and * Since .)(such that *)()(:

mapping onto one-to-one a exists Then there . tree

spanninganother Consider tree.spanning minimum a be *Let

tree.

spanning minimum unique has Then weights.enonnegativ

distinct have ),(graph connected of edges all Suppose

)()(

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Then, .most at lengths of piecessmaller into of edges all

partition torequired pointsSteiner ofnumber minimum the

denote )(Let .)(such that *)()(:



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minimum thefrom obtained is pointsSteiner ofnumber

minimum with spanning dsteinerize that theShow .most

at length ofeach piences into break them toedges itson points

Steiner some puttingby on treespanning a from obtained

treea is on treespanning dsteinerizeA number. positive

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Documents

Lecture 11 Overview Self-Reducibility. Overview on Greedy Algorithms