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
2. Lap lai qua trinh sau day cho den khi khong thg
thuc hien duoc niJa:
Chon b5 sung vao C mot dinh ke vdi moi dinh hien c6 trong C.
3. Ket qua dau ra la tap C."
5.5 Phan tick xdc suat cac thuat toan 345
De hinh dung ngay dugc hieu qua cua thuat toan Gruc, t a phat
bigu dinh ly saụ
D i n h ly 5.5.1 Thuat toan Gruc cho bai toan MAX-CLIQUE C6
cac hieu suat nhK sau:
(i) Rcruc = oo; (ii) Tren hau hit mdi dH kien G,
l o g n
trong do n la so dinh cua G;
0") ^ a r M c < 2 - •
Ta se t t o g budc chiing m i n h dinh ly naỵ Dieu (i) la hign nhign bdi v i , trgn do t h i dac biet G = (V, E) ma d6 t h i con cam sinh
G[y \ la day du va deg{ui) = 0, t a c6 RcrMciG) = n - 1 yh
do do RGTMC = CO - Digu (iii) dugc suy ra true tiep t i i dieu (ii) vh
dira theo dinh nghia ve hieu suat hau ch^c ch^n. N h u vay, t a chi phai chiing m i n h dieu (ii), n h u He qua 5.5.9 ci cuoi muc naỵ
Do RcrMci^) = ^GrM^^XG? ^ho ugn, d i CO dugc can t r e n t i e m
can cua 7 ? G r M c ( ^ ) i t a can phai t i m can tren cua OPTuc{G) vh cQ,n
dudi cua GrMc(G) doi vdi hau het mOi do t h i G.
Hay l u u y r^ng, trong ly thuyet do t h i , dai lugng OPTuc{G)
chinh la mot t h a m so dac t r u n g quan trong cua do t h i G, dugc ggi
la so clique {clique number), va con tUdng tU n h u mot t h a m so dac trung khac, do la sd doc lap {independence number). Cac tham so
nay da dugc khao sat nhieu doi vdi hau het c^c do t h i (xem, chang han, [7]).
Bay gid, dg danh gia hieu suat cua thuat toan Gruc, t a t i m each xdc dinh can tren cua OPTuc{G) \h. can dudi ciia GrMc(G)
346 Cac giai phap
• C a n t r e n ciia O P T M C ( G )
Can tren cua dai luong OPTMC{G) diioc irdc lirong doi vdi hau het
m8i dii kien G, b^ng each xae dinh mot so ko nao do sao cho trong
do t h i G khong ton tai bat cuf cUque nao vdi so dinh 16n hon hocic
b^ng ko, tire la OPTuc{G) < kọ Ta ky hieu:
Qn 1^ tap tat ca cac do t h i vdi tap dinh V gom n dinh dugc
dan nhan,
g„ = { G i | * = l , 2 , . . . , p } v d i p = 2 ( ^ ) ;
/C„ la tap tat ca cac do t h i con day du k dinh cua do t h i day
du vdi tap dinh V,
)Cr,,k = {Kj\j = l,2,...,q} v a i g = © ;
cjfc(G) la so tat ca cac k-clique, tile clique k dinh, cua do t h i G;
C k { n ) la gia t r i trung binh cua C k { G ) tren Qn, ti3c
-kin) = lZUck{G,).
Tap Qn duoc eoi nhu la tap tat ca cac du: kien "kich cQ" n cua
bai toan MAX-CLIQUE va, theo gia thiet, c6 phan bo xae suat deụ
Gia suf la mot bien ngSu nhien lay gia t r i cjfc(G), G e Q^, vdi
xae suat b^ng 1/2( 2 ) . K h i do ta eo
mn,k)=~k{n). B 6 de 5.5.2 Doi vdi moi \ <n, ta c6
CMng minh Doi vdi m5i do t h i Gj 6 ^„ va m6i do t h i day du k
dinh Kj e /C„^fe, ta dinh nghIa dai lugng xi^i, Kj) nhu sau:
X{G- K) — ^ ' ^ ( ^ j ) clique cua Gi\
\, trong trudng hdp ngUde laị
5.5 Phan tick xdc suat cac thuat todn 347
Theo dinh nghla Cfc(n) ta e6
^ i=l j = l ^ j = l i = l ^ j = l
trong do gn{Kj) la so tat ca cac Gj chiJa Kj, tilc tap dinh V{Kj) \h
fc-clique trong G^. Tiep theo, doi vdi bat ky Kj {l < j < q = Q)),
ta deu CO
9n{Kj) = 2&-í.).
Tit day suy ra dieu can phai ehiing minh. • B 6 d l 5.5.3 Gid sitG eQn- Khi do, vdi n du Idn, ta c6
1
Pr [OPTuciG) < 2 log n - 2 log log n + 4] > 1 -
l o g n '
Noi each khdc, doi vdi hau hit mSi dH ki$n G cua MAX-CLIQUE, OPTuciG) < 2 log n - 2 log log n + 4,
trong do n Id so dinh cua đ thj G.
Chvcng minh Ap dung bat dSng thiie Markov ngu tren vdi s = log n
cho dai lugng n g l u nhien ^n,k, ciing tiic la cho Cfc(G), ta thu dugc
1
Pr Ck{G)<c^{n) logn > 1 -
l o g n '
Tiep tuc, d6i vdi mgi k> ko vdi /co = 2 log n - 2 log log n + 4, theo B 6 de 5.5.2 va b^ng each danh gia tinh te ta c6 c}^{n) l o g n < 1 ,
khi n du Idn. T i l day suy ra r^ng, doi vdi mgi A; > A;o, P r k ( G ) = 0 ] > l -
logn 1 , khi n —y 00.
Dieu nay c6 nghia \h trong hau het m6i d6 t h i G khdng ton tai
bat eil clique nao vdi so dinh nhieu hon hoSc b^ng ko, cho nen
3 4 8 C a c g i a i phap
• C a n dvldi c u a Gruc{G)
Theo dinh nghia, gia t r i Gruc{G) so dinh cua clique C mk thuat toan Gruc t i m duoc k h i tinh toan tren diJ kien G cua bai toan M A X - C L I Q U E. Qua viec mo ta nhu tren ve thuat toan Gruc ta nhan thay r^ng, GruciG) Idn hon so Ian thuat toan Gruc thuc hien Budc lap 2. Bdi vay, d i danh gia Gruc{G), t a can khao sat dieu kien dam bao cho Budc 2 dugc thuc hien. Dau tien, de tien t r i n h bay, t a dinh nghia khai niem saụ
Sao (star), mot khai niem quen thuoc cua ly thuyet do t h i , la mot cay chiia duy nhat mot dinh vdi bac Idn hon 1, ngoai t r i i cay vdi 2 dinh. D i n h bac Idn hon 1 ay duoc goi la tam cua saọ M o t sao dugc goi 1^ C-sao (C-star), neu tap tit ca c^c dinh khac t a m ciia no la C. M g t C-sao t a m u dugc ky hi$u bdi S^{u) hay bdi 5^, k h i khong can b i i u lo t a m cua nọ
•
Gia sii G \h mCt do t h i bat ky vdi tap dinh V,\V\=n,\h. cho m6t tap con C cV, \C\ k. Ta ky hieu:
s^(G) Ik s6 tdt ca cdc C-sao cua do t h i G; s^(n) la gia t r i t r u n g binh cua s^{G) tren
Ik tap t a t ca cac C-sao cua do t h i day du vdi tap dinh V, S^ = {Sf\j = l,2,...,s} vdis = {n-k).
Gia sijt Cn,c la mot bien ngSu nhien lay gia t r i s ^ ( G ) , G 6 Gn,
vdi xac suat bang 1/2(2). K h i do, vdi p = 2(2), ta c6
i=i
B 6 de 5.5.4 Doi vdi mgi \ k < n, ta c6
E(Cn,c) = • :
5.5 Phan tick xdc suat cdc thuat toan 3 4 9 Chicng minh Tuong t u nhu B6 d^ 5.5.2, t a c6
trong d6p = 2 ( 2 ) , s = (n - k) va U{Sf) Ih s6 tkt ca cac d6 t h i
Gi e Gn mh chila Sf, tUc nhan Sf lam C-saọ Tiep theo, doi vdi
bat ky Sf e {1 < J < s), ta dg dang tinh dugc fn{Sf):
T\i day suy ra dieu can phai chiing minh. •
Dat ki = log n - 2 log log n, ta c6 ket luan sau daỵ
B o de 5.5.5 Dot vdi mSi tQp con bat ky cho trUdc'C C V vdi k dinh ma 1 < k < ki, khi n du Idn ta c6
Chitng minh Theo dinh nghia:
Var(C„,c) = E(C^,e) " ( E ( C n , c ) ) ' .
Bay gid t a danh gia dai lugng ^^i vdi m o i tap bat ky cho trudc C gdm k dinh ma 1 < A; < Jtị
Doi vdi mdi cap C-sao S^{u) va S^{ú) thuoc S^, ta ky hieu
f^^\s^{u), S^{ú)) la so tat ca cac do t h i Gi e Qn mh nhan ca hai
S^íu) va S^{ú) lam cac C-sao vdi tam u va ú tuong irng. B^ng each tach biet hai trudng hgp khi u = ú va khi u 7^ ú, ta dg dang tinh dugc / f \ 5 ^ ( ' u ) , 5 ^ ( u ' ) ) nhu sau: