Lecture 21 Tintin for Ai Az plane 2 finish arborescence Evaluations 2 matroid union Final May 24 a iz can access unitil Maps 9A once opened 3 hrs to finish practice final Open notes Spanning tree game Given graph G players alternate 1 Pl cuts an edge 2 P2 fixes some remaining edge P I can't cut fixed edges Pz cant fix cut edges P1 wins if graph becomes disconnected eg_P1 win Az plays bad
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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