Lecture 21 Tintinfor Ai Az

plane 2 finisharborescence

Evaluations 2 matroid union

Final May24 a iz can access unitil Maps9A

once opened 3 hrs tofinishpractice final Open notes

Spanning tree gameGiven graph G players alternate1 Pl cuts anedge

2 P2 fixes some remainingedge

P I can't cut fixededges Pz cant fix

cut edges

P1 wins if graphbecomes disconnected

eg_P1 win Az plays bad

pil

P2

RecadP2 wins if A 72 disjoint spanniytnees

in G

P2 uses thespannibtreesto maintain

connectivity

q f B I partition Vi Vpof v w

edges w endpoints indifferent Vi C 2 P l

T

sci upgo.im

P1 always plays edges from

d V Vp at the end P2

can save p 1 edges four 84 Vp

U Vp can't be connected O3

O_O

Today withmatroid union show u I

OkA Ee T B

i.e pz wins iff 32 disjt spanningtrees in G

Matroid UnionLee M CE I matroid

Recall dual matroid M CE I't

X EE EK contains abase of M

EI I f M MG for G fat

I't I 9 I

i.e subgraphs s t complement contains

a spanky tree If

theorem The dual matroid M't

is a matroid with rank function

rmx.cn7 lxltrmCElx rmCETproof one way Use

Fact Can define a matroid

using properties of rank functionif a function r zE N

a

is i imw

Then Mr LE II

I L S E E MS Is I

is no matroid w rank function r

Thus theorem follows from

A m is Mr for r rm

Largest elf of I in X has

cardinality rn X

B Rmt satisfies RO R l R2

A B left as exercise A

Eg disjointspauniy trees

G has 2 disjoint spacingtrees max ISI Iv I I

S C InT b c F spanning

MG7M

tree S whose

and L.C.IS algo complementcontains a

finds the trees spanningthe

min Max characterisation6

moreover matroidintersection

theorem

theorem G has two

disjoint spamnis trees

tf partitions V Vp of V

18 Vi NDI 32 p DEmine

G isi

Proof Assumeconnected elsetrivial

we onlyshowis exercise ra

Plain use Mitheorem forC matroid intersection therein

M M G NE Ecgets

whosecomplementcontainsbaseofM

Let n Iv I

G has 2 edge digitsparing trees

Max 1st a 1S EIRIK

rmCF n K Fe

C.c s in V F

Minear

MatroidithearernMax

S c Inl fineFuG

t rm EM

Recall we may assume U is closed

in M

padelfinFans cute ranked

i e U spanks for M MG u

is a union of subgraphs induced

by its c C s

e.ae U not closed a closed

i

o 088 s ofE

E vie

D finerainstormEM

CloseduinMJ A

ie f Ef

EE iY

amedntitfeu.aiu

min ut I t Isan vpHXp

V Vpcc's of U

by assumption 18Vi Up 132p D

the alcove is 3 nt I TypD 2

PE

a 2disjt spanningtrees I

midintersection but used itto solve union like problemgeneralizes

generalizes

CGeneralmatunionLet M CE Ic Mr LE I

matroids

Def The matroiduuion

M U Mz E I

I X VY X C I Y c Iz

Careful If I VI z

Theorem M V Ma is a matroid

has rank function

rm.umdstnuriegflslultrm.lurmdM3

Consequencescan effeciety

decide ifthere are two dijoint bases

of Mi MzBi Bz

ble this happens

largest indep sit in MNMz

has size rn E tradesizeofahaseinmhsizea.semm

can decide my greedyalgCan we find Bc Bz A littlemorework to get it from Bev BzIn fact M U UM u

also a matroidcan solve matroid partition

problem of deceits if

E B b UBk basesofM B M k

Proof x QQ e 00Ts Yz

Partly M UM is a matroidP 12 easy PLLet x e EI NICKI

and X X V Xz Y Y Ok

Xi Ki c Ii disjoint usedownward

closed

maybe empty Ryoptwanintersections

Need to show Jee YuSt Xt e c I

Assume away choices of Xi Miours maximizeHim It 1 4421

Since Tel Xl assume

tell X l switch 22 32

if necessary

3 e EY Xi s t X te EI

ee X z or else x c XiteXz Xz e

increases 1 119,1 1 2 A 421e Yz by disjointners off

ee Y IX's EnduroX t ee I

PartI Rank function

rm.umfst runeigfiskltrm.lu trainx E

Il

IL XiuSEI

E clear snu X 0 2

151 15141 Isna AjxitIiMfi term lu

e Isnt t fufu trmdu

matroid intersectiontheoremFor 3 usee

not

First prone for 5 E proof is eats

sit X z te EIX Z

for others folllous byrestricting Mds Mds We

Xx X e

Let X base of M U M z x zte

X X j x zneed to show

IX I Minh l EUI t

why rm.cn

May assume rmdXz fµdEMm M

add to XL remove from X I

1 1 Ix I tried E

f x OX this is the.fiefotMz

b

Then XitI and XIEIE because

Efx contains base Xz C I 2 base in

72I.e X C I NII int

matrord intersectiontheorem for M ME

Funny 1 1

xm.EI.rslXiltrmzCED

EE.n zlXil trmzLE

using min rm.lu trmdEM trmdEM I T WEE fespression

min rm.lu tIElultrmzfujmEU EE rµzlE trmzEJ

miferm.lu trouzcultleml

endparID

minust subarodulanuniform.umds