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

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

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

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

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

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

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

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