grup tree - department of computer sciencemdinitz/classes/approx...grup steiner tree input h v el...
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Grup Steiner Tree
Input h V El
edge c sts c C Reovertex r root
K groups 91,94 gg e eau4giEVFeasible Tree TEE s t fieck Ivegi where The
an r v p th
objective min CCD ele del
lyToday h a tree cast on trees
WLOL all terminals are leaves
Thy let cover is a special case of LIT on trees
PI Let U.S instance at set cover
µ view si HS its
I se i Sm el 1 Keef
let car is valid hst solution
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a y hit is a set cover
core NP hard to approximate better than rlly n
tha Halperin Kranthgame 03 A constant e O
Np hard to approximate GIT on trees better than
log n
TheChag Knievel Ravi 00 There is an OClasalog k
approximation algorithm E LST on trees
ILPrelaxationr
mine cCle Xe
g t E Xe 21 V icCKBV sevi.t.rs ginseels 5
0 Exe E 1
The ILP add xeelo I constraints is an exactformulation
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Solve Lp
Ellipsoid Separation oracle
Separate Liven is there an S separating r
from gi with Exec 1effs J
min c t
T s
E E.ae ti
Too
Equivalent interpretations of LP by max flow min c H
1 find x s t every c t separating r from Si
ha I 1 total across
2 find x i t if we think of as capacities
we can send 1 fro from r fo g
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Rounding Can't ra d each edge independently
I t t L
feeu
r s e e 9 yulL Z Z Z L
v
Prlr connected to g EY
r KI
Inflation like set cover not helpful yaMioDEI Let ple parent edge of e µ
Lenny Xue I Xe He in any optimal solution
Pf sketch
Olivia when think of capacities
r
1 If Xpce C Xe then carat
Mel utilize the capacity at e
He Decrease Xe to Xue getA11 A better solution
x optimal
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Basicity GKR Raiding
Solve LP to get x
for each ee fma k e with probability In
Xpce I f e ha no
parent edge line L f on r mark with pah Xe
for each ec.fi include in T if naked and all ancestors
marked
Lemmy f Ce int Xe
Eirunts I I i
Xe
corollary CCALL ELP
CCALL C fee del Ileet efe.de xe LP
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The Vital
P Lg connected t r in TI Z ftp.T
first use them to get Olly a leg 14 approx
Run basic alg 0Closs kik times take union
If gi not connected to r add cheapest r gi path PN te ccPi E OPT
feasible with probability 1
f Cgi not connected to r in union E
E Ct ITIt
EeO
s t
EcceALLI E fugal H Let it tic evilc 0144414 opt II Tc OPTc OClagulag H OPT
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Rest of class proof of thm
The Vital
P Lg connected t r int I ftp.T
Let g g be a group
et FAIL be event that r not connected to g
Lemme If x'e E Xe te then 4P C FAIL using 2 Prc FAIL using
Pf sketch i
One edge at a time induction
Single edge e with axe all other I have x'en x
u T I letree rooted at u
e Nothing outside of T change
an P Prc fail to connect Ting ta w
y p Pa f Cfa L to connect Tang to 2
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using DPrc fail to connect Tng to u
Ctu talk It A Ct It rm pr a
Pile not added free art marked failbelow e p.ee ael naked Ibelow ee
I xe.ltp.l xe.CI pal xe.xeclte.luXe
increase as Xe decreases
So we can use smaller x in analysis
if f Cg connected to r in This 2 t.gg then
Mcg connected to r in Tring x 2 1
create x
1 Remove all leaves out in 9 all unnecessary edges
2 Reduce value until minimally feasible
exactly 1 flow can be sent from r togmin r g c t 11
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