CSE 565 , lecture 14, 10/13/2021
projection problem
Description ④ -8
-30B€ b
G=N
PW )
: profit of v.
t.tt
④→⑤
-7 10
DI Hea side 🙂 Vi EV is feasible , if V1 satisfies :
if u C- V , and Lui V1 C- E , then UE V1 .
Input : G-=/VIE) . plusV-vc-V-uut.IOa feasible V1 c- V sit . ¥yPM is maximized .
Algorithm : reduce into the max- flow problem .
VIVI
step create G-= IVIE) . Cle ) He c- E , sit,
1-*
¥÷¥÷É¥¥÷:*¥ .
step find f-
*
of this network .
an s-t cut 15×1-1*1
step return 1-
* 1ft}
Fa -1*1 It} is feasible .
proof :
1-*
⇒ 45×1-1*1 _→ , a contradiction .
FAI ¥g*yµPM
is maximized .
”
☒
*
s*
clS¥T*)=¥g¥P
‘” t ¥
,#
1- PWD
=/I PM -[ PH – ¥g±PMv c- 540T¥ VE -1¥
=¥µ,>0PM
– ( ¥¥PMt¥g±Pm)
= c- I
VE-1*1 /ty
P”” ☒ .
Cawnstant)
Baseb.at/Fliminationproblemourrentswringboard- schedule of remaining games .
I w ‘ (Ti,Tj)✗XijgamÉ
T2 WZ
: :
Tn Wa
In teams I # wins)
Question :
is IT -☒ill possible to win the 1st place ?
I #wins 1-* gets is among thehighest.IT
Is T2 -13 -14 TsE×aM
#wing * a 41 43 39 –
M= 43 42 42 43
scheduling text I :O
(1–74)×-2 2 :O
¥9
(Tz , -13) ✗ 3
Tc={Tz , -13, -14} . ( Tz, Ta) ✗ 1 I :O
(-13,1-4) ✗ 1 I :O
(-13,1-5) ✗ 2 .
Preprocessing : assume Ti wins all remaining games
⇒ use m to denote the # total wins) of -1 ,
Question : Is there awntigurationofallremaininggamess.fi#Wittt-fi=m.V-i >2. ?
Algorithm : reduce into Max – flow problem .
step create G- = IV.E) Cle) Yet E. sit .
m-Wi
i.
2
s-% 4
(games ) (teams
skp find f-
*
.
15%-1*1
.
step 3 : It 11*1 = ¥gzXij ,
then answer ” yes
”
otherwise
, answer
”
no
”
.
Proof of correctness :
11-+1 = ,§⇒xij ⇐ such a Intention exists ☒
certificate-1,75¥ :-b .
‘
‘
: Tn } that shows -11 is eliminated :
Hi , -1J ) c- To
✗ it > I Lm
-Wi .
TIETC
Q : Does a Tc always exist when 11–4 < §g⇒Xij ?
A : Yes .
constructively f
'
4551-7-6
m-Wi
:*
¥É÷F ⇒
Claim : If ITi. Tj) Est , then Ti c- stand Tj c- s*,
±t¥i÷÷*
⇒ 45×1-1*1=0 , a contradiction ☒
Clavin : If Ti , -j c- s*, then (Ti-Tj) c- s* .
proofs
,
1-
*
By moving IT i.Tjlto 51
that edge is excluded from cut-edges
-
⇒ CIST-1*1 won't be minimum , a contradiction . ☒
claim: :TofTiES*} isauria . it HHi§⇒×ij .
ie' (¥g%×ij
> ¥ M
– Wi
proof : 11*1-+1511-+4
I
=¥Ém
-Witt i¥z×ij – (Ti , ETC
sine 11*1 < ¥>.< Xij ⇒ ¥TEKXY " > ¥ M
– Wi ☒