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. Kiim tra xem lieu trong danh sach thu duoc c
hay khong clique vdi i t nhat K dinh. Neu c6 t h i chap nhan; nguoc lai, bdc bọ"
Do thuat tôn TIAS c6 do phiJc tap th5i gian 0[n-m-ri] n h u da neu tren, nen thuat toan TIASct^ cung c6 do phiic tap tuong t u 0[n-m'-fi], trong do m' = (") — m va // = IJ-{G) la so tat ca cac clique toi dai cua G. Dg chufng to TIASCN la thuat toan thdi gian trung binh da thiic cho cac bai toan CLIQUEN(ri) va CLIQUEN(n)^,
ta chi can chting m i n h rang gid tri trung bmh cua so t a t ca cac
clique t o i dai //(G) tren tap Gn,N (va tUOng ling tren tap Qn^i^) la
da thiic theo n, trong do Qn,N la tap tat ca cac do t h i n dinh ma
CO khong qua N{n) canh (va Q^j^ la tap tat ca cac ^6 t h i n dinh
mh CO dung N{n) canh). R6 rang
Theo gia thiet, tren cac tap Qn,N va Q^f^ c6 phan bo xac suat dgụ Dat a = \g^!^\a 6 = |^„,/v|. G i a sijt Cfc(G), c ( G ) va / i ( G ) tuong
utng la so t a t ca cac A;-clique, so tat ca cac clique va so t a t ca cac clique t o i dai cua do t h i G . Ta xet cac dai ludng ngau nhien sau:
en,r^,k = Ck{G), e,;v = c ( G ) , M U = M^ ) ' trong do G e g^,;v, vdi xac suat la l/o;
Cn,N,fc = Ck{G), ^n,N = c ( G ) , Hn,N = A*(G), trOU g do G G gn,N,
v6i xdc s u i t la 1/b.
5.5 Phan tich xdc suat cdc thuat toan 3 6 6
B o de 5.5.10 Doi vdi mgi l<k<n vd 0< N < (^), ta cd
Chvcng minh Hokn tokn tuong tU n h u B6 de 5.5.2. •
B6 de 5.5.11 Doi vdi mgi 1 < k <n vd 0 < N < n ^ - ^ trong
do 0 < c < 2, khi n du Idn ta cd
Chiing minh Theo B5 de 5.5.10 ta c6
© ( G ) -1 ) ( 0- © + !)- I®
(5)
\G)/
< ^{4+£)fc/4-£fcV4^
do 0 > k h i n du lan. Dat f{x) = (4 + e)x/4 - exV4, ta c6
V i the, doi v6i moi 1 < A; < n , ta t h u dUdc
• T i l b6 de nay t a de dang di den nhflng ket luan sau daỵ