I. Recouvrement d`ensemble est NP

n
n Ui, Uj
vi,j Ep, Eq
cp,q
a1, . . . , amb1, . . . , bm
aibj
p
q=n−p
A, B, C, D a, b, c, d
π:A7→
a, B 7→ b, C 7→ c, D 7→ d A d
ES=
{Si}1≤i≤kE Siw(Si)
J⊆ {1, . . . , n} ∪j∈JSjE
Pj∈Jw(Sj)
N P
W
N P
G= (V, E)
k X ⊆V k
(u, v)∈E u ∈X v ∈X
J
X|X|X\Y X Y

F← ∅ J← ∅
F6=E
c(Si) = w(Si)/|Si\F|i= 1, . . . , n
i c(Si)
F←F∪SiJ←J∪ {i}
J
(Ti)1≤i≤k
W0Si
Siw(Si)
|Si|≤W0 n
w(T1)
|T1|≤W0 n
x1, . . . , xnE F
TixjF
c(Ti)≤W0/(n−j+ 1)
HnHn= 1 + 1/2 + . . . + 1/n
n+ 1 E={1,2, . . . , n}Si={i}1≤i≤n Sn+1 =E
ε > 0
Hn/(1 + ε)
Σ Σ f
`(f)g f f0f00 f=f0gf00
S={f1, . . . , fn}f fi
fifjk fik
fjk σijk ni+nj−k
finj−k fjninjfifj
S={f1=abk, f2=bkc, f3=bk+1}σ1,2,k f
fifjS
k k fifjS σi,j,k
S
P S ={f1, . . . , fn}Q
Q E =S Su
u u S σijk (fi, fj)
k Suu S w(Su) = `(u)
u
Sθ1, . . . , SθpQ W0
W0P
f P S
fififjf
i < j fif
1 = b1< e1< b2< e2<· · · < bt< et=n i = 1, . . . , t
bi=ei−1+ 1 eiS fbi
kifbifeiπi=σbieikiSπ1, . . . , Sπt
Q
πiπi+2 f f
Pi`(πi)≤2`(f)
2Hnf fi