I Dinh ly 5.4.2 Khi P^ NP, moi thuat todn xap xi thdi gian da thiic giai bai todn M A X C L Q U E deu co hi^u suat tuyet doi bang 0 0
350 Cdc giaiphap
Do so c^c cap S^{ú)) vdi u = ú Ih. n — k vk so cac cSp
c6n lai \a {n - k){n - k - I). Cho nen, tuong t u n h u tren, t a c6
= - A:)2(=H + (n - fc)(n - A; - 1)2(^H'^} 1 ( n - f c ) ^ (n - fc)(n - fc - 1) 2k 2^= (n - - 1) < (n — fc) (n — /c) •(E(C«,c))^
trong do bat dang thu:c cuoi ciing duqc suy ra i\i gia thiet k < ki vdi /ci = log n - 2 log log n va k h i n du Idn.
Vay la
I
Var(C„,c) = EiCc) - ( E ( C n , c ) ) ' < ^ 4 " ( E ( C n , c ) ) ' ,
log n
dieu ma t a can phai chiing m i n h . •
B 6 de 5.5.6 Gid svC G 6 Qn- Doi vdi moi tap con bat ky cho trade C C V vdi k dinh ma 1 < A; < fci = log n - 2 log log n, khi n du Idn ta c6
2 loglog^ n
Pr s^(G) > 1 > 1 -
log^ n
N6i cdch khdc, trong hau hit moi do thi G = {V, E), doi vdi moi tap con hdtky C CV ma \ \C\ log \V\ 2 log log \V\, ton tQ.i
it nhdt mot dinh thuoc V \C va ke vdi tat cd cdc dinh cua C.
5.5 Phan tich xdc suat cdc thuat todn 3 5 1
ChUng minh A p dung bat dang thiic Chebyshev vdi t = '^^^"•g)
log log n
cho dai lugng n g i u nhien Cn,c va theo Bo de 5 . 5 . 5 , t a t h u dugc
Pr[|Cn,c - E(C„,c)| > E(<^„,c)/loglogn < 2 log log^ n log^ n
Tiep tuc, theo B Q de 5 . 5 . 4 dinh nghia C„_c, t a c6
Pr 1 - 1 \n-k V log log n / 2*= > 1 - 2 log log^ 71 log^ n Do (1 - i ^ ) " ^ > 0 k h i A; < l o g n - 21oglogn, nen P r [ s ^ ( G ) > l l > l - 2 log log^ n log^ n
T i t b6 de viia duoc chiing m i n h t a c6 ngay menh de saụ
•
B 6 de 5.5.7 Gid sit G E Q^. Khi do, doi vdi hat ky ho C^^ki,
CnM = [C^''^ I C^'"^ ^ ^ va |C('=)| = A;, doi vdi m5i 1 < A; < A;i},
trong do \V\ n vd ki = l o g n — 2 log l o g n , ta cd
3 loglog^ n Pr Sc(k){G) > 1, doi vdi moi > 1 -
l o g n
•
Quay t r d lai vdi thuat toan Gruc ma t a da mo t a cho bai toan
M A X - C L I Q U E . B 6 de n^y cho ta thay rang, tren hau het moi du:
kign G = (V, E) ciia bai toan, ttrc mot do t h i gdm n dinh, so Ian thuat toan Gruc thuc Men Budc lap 2 it nhat la l o g n - 2 log l o g n .
Cu thg, t a i Ian lap t h i i A;(l < A; < log n - 2 log log n), doi vdi A;-clique
C^^) t i m duoc trudc day, trong do t h i G ton t a i dinh Uk+i G K \C('=)
ke vdi moi dinh cua C^^\d do ta xac dinh duoc {k + l)-clique
tiep theo C^^+i) = C^'^) U {uk+i). V i vay, lien quan den gia t r i cua
352 Cac giai phap
He qua 5.5.8 Gid siC G e Gn- KM do
Pr \GrMc{G) > log n-2 log log n] > 1 - 3 log log^ n log n
Noi each khdc, tren hdu hit mdi dii kien G cua M A X - C L I Q U E , thudt todn GTMC t^'m duac nghiem vdi gid trj GruciG) thoa man:
Gruc{G) > log n - 2 log log n,
trong do n la so dinh cua do thi G. •
• Cuoi cung, do °GJUC{G) = ^ruc 1^ hieu suat cua thuat toan
Gruc tren G, cho nen tii BO dg 5.5.3 va He qua 5.5.8 ta di dgn
khang dinh dieu (ii) cua Dinh ly 5.5.1.
He qua 5.5.9 Gid sv£ G e Gn- Khi do 3 log log n
Pr 7?GrMc(G')<2 +
logn > 1 -
4 loglog^ n
log n
Noi each khdc, tren hdu hit mdi dii kien G cua M A X - C L I Q U E , thudt todn Gruc c6 hieu suat RcrMci^) ^^^^ man:
3 log logn I
7?GrMc(<^)<2 +
trong do ri la sd dinh cua do thi G.
logn
Nhu vay, ve tinh hiJu hieu cua thuat toan Gruc ta c6 doi dieu
ket luan sau daỵ Giong nhir moi thuat toan khac cho bai toan M A X -
C L I Q U E, thuat toan Gruc c6 hieu suat tuyet doi RGTMC = ô! nghIa
1^ trong trudng hop xau nhat, thuat toan Gruc tim duoc nghiem
khac xa vdi nghiem toi irụ Tuy nhien, bang each phan tich xac
suat, ta chi ra rang thuat toan Gruc c6 hieu suat hau chac chan
^ G r M c - 2; nghIa la tren hau het moi du: kien bai toan, thuat toan
Gruc cho ta nghiem vdi gia tri khong Idn hon khoang mot nufa gia
tri cua nghiem toi uụ
5.5 Phdn tich xdc suat cac thuat todn 353
5.5.2 Do phufc tap thdi gian trung binh da thiic
Nhu tren da noi, do khong ton tai thuat toan thdi gian da thiic giai cac bai toan NP-day du hoac NP-kho noi chung (khi thiia nhan
P ^ NP), nen viec tim kiem thuat toan vdi do phiJc tap thdi gian
trung binh da thufc cho cac bai toan ay la mot giai phap dung d^n
va r4t CO y nghiạ Mot vai bai toan NP-day du da dtroc chilng minh
la giai duoc trong thdi gian trung binh da thilc. Sau day ta gidi
thieu mot ho cac bai toan con NP-day du cua bai toan CLIQUE vk
chilng to rkng chung giai duoc nhu vaỵ
Cho ham so : N —> N ma 0 < A^(7i) < n^'^ va tinh duoc
trong thdi gian da thiic, trong do e la mot h^ng so hiiu t i thoa man 0 < e < 2. Ta xet cac bai toan C L I Q U E N( n ) va C L I Q U E N( n ) % dirge khao sat trong [25] va duac phat bieu nhu sau:
C L I Q U E N( n )
DU ki$n: Cho mOt do thi G = {V, E) vdi |E| < Â(|V^|) va mOt so nguyen duong K <\V .
Cau hoi: Phai chang trong G c6 clique vdi it nhat K dinh?
C L I Q U E N( n ) ^ khac C L I Q U E N( n ) d cho la chi bao gom nhiing dii
kien (G, K) ma G = (V, E) vdi \E\ N{\V\).
Vdi gia thiet ran^ 0 < N{n) < n^'' (0 < e < 2), cac bai toan
C L I Q U E N(T I) va C L I Q U E N( n ) ^ duoc chuTng minh la NP-day dụ
Bay gid ta xay dung thuat t o a n m ^ c N vdi do phufc tap thdi
gian trung binh da thilc cho cac bai toan nay, chu yeu dua theo
mot thuat toan ma ta goi la TIAS, do b6n tac gia Tsukiyama, Ide,
Ariyoshi va Shirakawa de xuat trong [32] d i liet ke tat ca cdc tap
doc lap toi dai (maximal independent sets) cua do thi G. Thuat toan TIAS duoc chiing to la cd do phiic tap thdi gian 0\n-m-ri\, trong do n la so dinh cua do thi G, m la so canh cua G, va 77 = 77(G) la 50 tat cd ode tap dgc Idp toi d^i cua G.
354 Cdc giai phap
Thuat toan TIAScu cho ca hai bai toan CLIQUEN(n) va
CLIQUEN(n)^ diroc mo ta sd luge nhir sau:
m5cN = "Tren dau vao {G, K), trong do G = {V, E) la do t h i
vdi \E\ N{\V\) (hay v6i \E\ N{\V\), doi v6i bki
toan CLIQUEN(n)^), va la so nguyen diiong <