Lecture 10 Matroid

Independent System

• Consider a finite set S and a collection C of subsets of S. (S,C) is called an independent system if

CACBBA ,

i.e., it is hereditary.

Each subset in C is called an independent set.

Matroid

.}{:\

|||| and ,

:property exchange thesatisfiesit if

matroid a is ),( systemt independenAn

CxBBAx

BACBA

CS

Matric Matroid

ce.independen

linearly for fact known wella ispropery exchange The

matroid. a is ),(

Then . of subsetst independenlinearly all of collection the

and of vectorsrow ofset thebe Let .matrix aConsider

CS

SC

MSM

Graphic Matroid

.}{ implies

which ),,( of components connected twoconnected edge

an has Hence, .components connected |||| has ),(

and components connected |||| has ),( that Note

. with ,consider property, exchange show To

matroid. a is ),(Then . ofsubgraph acyclic

an induces which ofeach sets edge all of collection the

and Let ).,(graph d)(undirecte aConsider

CeB

BVe

ABVBV

AVAV

|B||A|CBA

CSMG

C

ESEVG

G

Extension

.in extension no has if

in maximal is set t independenan ,subset any For

extension. no hasit if maximal isset t independenAn

t.independen is }{ and if

set t independenan ofextension an called is element An

FI

FFISF

xIIx

Ix

Maximal Independent Set

).()(,any for ifonly and

if matroid a is ),( systemt independenAn

} of subsetst independen

maximal allover is | |{| min)(

} | |{| max)(

define ,For

).,( systemt independenan Consider

FvFuSF

CS

F

AAFv

CIIFFu

SF

CS

Theorem

Proof

.in extension an has so , ofset

t independen maximal a becannot Hence, .|||| size has

ofset t independen maximalevery Then .Set

.|||| with and setsindepent woConsider t )(

. of maximality

theingcontradict , ofextension an contains then

|,||| if , and setst independen maximal For two )(

FBF

BBA

FBAF

BABA

B

BA

BABA

Basis

tree.spanning a is ),(

ifonly and if basis a is ),,( matroid graphic aIn

basis. a

called is ofset t independen maximalevery ,),( matroid aIn

AS

CACSM

SCS

G

Weighted Independent System

basis. a isset which independetweight

maximum a is therematroid, weightedaIn

system.t independen

weighteda called is :function

enonnegativ a with ),( systemt independenAn

RSc

CS

Minimum Spanning Tree

edges. ofnumber fixed a contains which treespanning a i.e., base,

a isset t independen weight maximumevery because is This

).(max where),()(*weight

with matriod graphic in theset t independen weight maximum

thefind toequivalent is treespanning minimum thecomputing

,: weight edge with ),(graph connected aFor

maxmax ecceccec

M

REcEVG

Ee

G

Greedy Algorithm MAX

.output

};{hen t

}{ if

do to1for

;

).()()(

ordering into in elements allSort

21

A

xAA

CxA

ni

A

xcxcxc

S

i

i

n

Theorem

.in

set t independen maximal of size minimum theis)( and

in set t independen of size maximum theis )( where

)(

)(max

)(

*)(

Then solution. optimalan be *Let

Algorithm.Greedy by the obtained beLet

system.t independen weightedaConsider

F

Fv

FFu

Fv

Fu

Ac

Ac

A

A

SFG

G

Proof

)(|*|))()((|*|

*)(

Similarly,

)(||))()((||

|)||)(|(||)(

)(

}. ..., ,{ Denote

1

1

1

1

1

1

2

111

1

nn

n

i

iii

nGn

n

i

iiGi

n

i

GiGiiG

G

ii

xcASxcxcAS

Ac

xcASxcxcAS

ASASxcASxc

Ac

xxS

. ofset t independen

maximal a is is, that ),(||

show only to need We).(|*| that Note

.||

|*|max

)(

*)(

Therefore,

1

i

GiiGi

ii

Gi

i

niG

S

ASSvAS

SuAS

AS

AS

Ac

Ac

. choosingnot of rule theingcontradict

t,independen is }{)( So, t.independen

is }{)(such that \ a is Then there

in maximalnot is set t independen Suppose

1

.

j

jGj

jGiGij

iGi

x

xAS

xASASx

EAS

About Matriod

Theorem An independent system (S,C) is a matroid iff for any cost function c( ), the greedy algorithm MAX gives a maximum solution.

Proof. (=>)

solution. optimal gives MAX Therefore,

.any for )()( matroid, a is ),(When SFFvFuCS

Next, we show (<=).

Sufficiency

optimal.not is

which, produce willalgorithmgreedy The ).()(

satisfy number to positive smallly sufficient a is where

)(\ if

\ if

if

0

1

1

)(

Define

.|J||I| with and setst independen maximal two

has such that exists Then there matroid. at isn'

),( systemt independen suppose ion,contradictFor

IJcIc

JISe

IJe

Ie

ec

JI

FSF

CS

A Task Scheduling Problem

Unit-time Task Scheduling

deadline. missedfor incurredpenalty totalthe

minimizes that me within timachine aon for schedle a find

. by time finishednot is task if incurred is penalty a

such that ,,...,, penaltiesor weightsenonnegativ ofset a

and ; by timefinish tosupposed is task and 1

satisfies each such that ,,...,, deadlinesinteger ofset a

tasks;time-unit of },...,2,1{set a

11

21

nS

diw

wwwn

dind

ddddn

nnS

ii

n

ii

in

Input

Output

Independence

late. is task no then deadlines,

ingnondecreas oforder in scheduled are in tasks theIf )(

Trivial. )(

.|}|{|)( where)( ,,...,2,1

any for ifonly and ift independen is tasksof set A

penalty. without tasksthese

for schedule a exists thereift independen is tasksofset A

A

tdAiANtANnt

A

itt

Lemma

Proof.

Matroid

.1for )(1)(}){(

and 1for )(}){(Then

}.1|\{ Choose .1for )()(

and Then ).()(such that largest thebe Let

.|||| with and setst independen woConsider t (exchange)

Trivial. )(herditary

matroid. a is ),(Then . of subsetst independen all of

set the and deadlines with taskstime-unit ofset a be Let

ntktBNANxAN

kttANxAN

kdABixntkBNAN

nkBNANtk

BABA

CSS

CS

ttt

tt

itt

tt

Theorem

Proof

.0 Possibly, k AB

10 2

K=?

?k AB

10 2

AB

10 2

AB

10 2?k

?k

Another Example of Matroid

matroid. a is ),(Then

}.|{

denote , fixedany For

).,( systemt independenan Consider

A

A

CS

BAEBC

SA

CS

Proof

.1||)()(

Hence, . somefor }{ formin is of

subsett independen maximalevery then , If

.||)()( Hence, . isch subset whi

t independen maximal unique has then , If

.Consider

FFvFu

AxxFF

FA

FFvFuF

FFA

SF

What we learnt in this lecture?

• What is matroid?.

• matric matroid and graphic matroid.

• Relationship between matroid and greedy algorithm.

Puzzle

matroid?

a induce subgraphs acyclic all do graph, directed aFor