. Bai toan VERTEX COVER Day \h mOt trong nhiing hh
Dinh ly 2.5.5 Neu Li dp ^2 thi Li"
Chitng minh Gia sfl Li \k ngon ngfl trgn bang chfl Sj (i=l,2), vh
/ : EJ —> Ê la phep quy dan thdi gian da thflc t f l Li den L2.
Khi do, doi vdi moi lu e EJ, ta c6 w 6 Li f{w) 6 L2 va do
<J6 w 0 L i < ^ f{w) ^ L2, tflc w 6 Li^ ^ f{w) e Dieu nay
196 Do phtic tap thdi gian
: ^
Bay gid, d6i vdi mot Idp phiic tap K cho trirdc, phdn bu cua no
duoc dinh nghia nhu sau:
co-K = {L^ I L e K } .
• Doi vdi Idp P. Dg thay rSng Idp P bat bi6n du6i tac dong
cua phep lay phan bii, tiic P = co-P. Ta chi can chiing to rS,ng,
neu L e P thi // e P. That vay, gia sii L duoc khing dinh bdi
may Turing tat dinh th5i gian da thufc Af. Khi do duoc khang dinh b5i may Turing tat dinh thdi gian da thdc M'^, nhan duoc tit may Af bang each dcn gian hoan dCi vai tr6 eua ckc trang thai "chap nhan" vk "bac bo".
• Doi vdi Idp NP. Mot cau hoi duoc dat ra kha tit nhien Ih: Lieu
Idp N P CO bat bien hay khong dudi tac dong cua phep lay phan biị
tile N P = co-NP. Khong dan gian nhit doi vdi Idp P, day la mot
trong nhiing van de cd ban va rat hoc bua cua ly thuyet do phutc tap tinh toan. Sau day ta ly giai phan nao tinh hoc bua cua v4n d i .
Cho L la mot ngon ngii tren bang chii S vh thuoc N P nhunj
khong thuoc P (khi thita nhan P ^ NP), chang han nhu ngon ngu:
NP-day diị Gia sut la may Turing khong tat dinh thdi gian d
thiic khang dinh L. Theo each lam nhu doi vdi ngon ngu: thuoc P,
dg khang dinh ngon ngu: bu L'^, ta thii xay dung mdy Turing khon ;
tat dinh thdi gian da thilc Â*^ tit mdy N hKng each hodn doi vai tro
cua cae trang thai "chap nhan" va "bae bo". Nhan thay r^ng, trf i
mdi tir vao w e T,*, hai cay tinh toan Ti^{w) vh TNC{W) gi6ng het
nhau ve hinh dang c6n ve noi dung (tiie hinh thai d ckc dinh) cui I
CO ban nhu nhau, ngoai trir eae hinh thai ket thiic d txlng cap dinh la
tuong ling cua chiing cd trang thai doi lap nhau, giiia "chap nhS
va "bac bo". Doi vdi cay tinh toan TN{W) tven tit vao bat ky i^'
cd ba kha nang xay ra: Thvc nhat - hoan toan "chap nhgn", khi t t
ca cac nhanh deu dan den hinh thai chap nhan (Â chap nhan ú);
Tha hai - "chap nhm" va 'Uc bo", khi cd nhanh dan den hinh t\^^ chap nhan vk cung c6 nhanh dan den hinh thai bdc bo (Â chaP
2.5 Cau true cua cdc Idp NP va co-NP
197
nhan w); ThU ba - hoan toan "bac bo", khi tat ca cac nhanh dgu dan den hinh thai bac bo (Â bac bo w). (Co le chi doi vdi cac ngon
nga thuoc P, kha nang thtt hai mdi cd the khong xay ra). Khi do
ba kha nang xay ra d6i vdi cay tinh toan TNC{W) tuong ilng la: ThU nhat - hoan toan "bac bo" {N^ bac bo w)\ hai - "bac bo" va "chap nhan" {N^ chap nhan w, theo Dinh nghia 1.2.3); ThU ba - hoan toan "chap nhan" {N"" chap nhan w). Do do may Ấ^ khong chi
chap nhan tat ca cac tit thuoc mk con chap nhan them nhieu tit khac dUdc chap nhan bdi N, tiic nhiing tit thudc L; nghia la ngon
ngii khong duoc khing dinh bdi may Turing khong tat dinh Á'^. Cho den nay vtn khong cd each nao xay dung may Turing khong
tit dinh thdi gian da thiic d l khang dinh ngon ngii bii cua bat cii
ngon ngii NP-day du nao cho trudc. Nhu vay, giai quyet van de
N P = co-NP bang "phiiong phap lua chon" (xay ditng may Turing
khong tat dinh) dang gap phai nhiing khd khan dang kg. Sau day ta trinh bay cu thg hon ve nhiing khd khan khi giai quyet van de nay b^ng "phuong phap kilm chiing".
Gia thijr N P = co-NP. Khi do, neu ta coi N P nhu Idp cac
ngon ngii duoc kigm ehiing nhanh theo nhiing bang chiing sue tich
(Dinh ly 2.3.5), thi co-NP at phai la Idp cac ngon ngii duoc kiem
chiing nhanh theo "nhiing chiing cii phan bac" siic tich, tilc "nhiing
bang chiing ngoai pham" doi vdi ngon ngii tuong ling trong N P va
vdi do dai da thiic. Nhiing bang chiing nhu vay doi vdi cac ngon
ngu: cua N P noi chung, trii nhiing ngon ngfl thuoc P, hau nhu la
khong thg tim kiem duoc. Viec khao sat d cuoi Muc 2.3.1, doi vdi t>ai toan ve ditdng di Hamilton trong do thi cd hudng, phan nao
chiing to nhan dinh naỵ Doi vdi nhieu bai toan khac trong Idp
NP, dac biet la doi vdi cac bai toan NP-day dti, ta cung c6 nhan
^et tuong tit, tiic khd cd thg chiing to rang cac bai toan bu ciia chiing thuQC N P bang each kigm chiing nhanh. Bdi vay, ding thiic
= co-NP khd ma chiing minh duoc b^ng phuong phap kigm
Do phufc tap thdi gian Bai toan bu cua bai toan SAT ducJc phat bieu nhu saụ
SAT
Da kien: Cho mot cong thilc Boole (j).
Cau hoi: Phai chSng (j) la khong thoa diroc, trie mau t h u l n ,
nghia \k khong mot phep gan t r i Boole nko thoa 0?
De thay r^ng, doi vdi bki toan SAT, bling chilng de khang dinh mot cong thiic Boole 0 la thoa duoc chi la mot day thuoc { 1 , 0 } ' " bao gom m ky t u 1 va 0, trong do m la so cac bien xuat hien trong
(j) con 1 va 0 la ky hieu cua nhflng gia t r i chan ly DUNG vk SAI tuong ling. The nhung, doi v6i bai toan SAT, b^ng chiing dg khang dinh cong thiic Boole 0 la mau t h u l n phai la day bao gom toan bo tap { 1 , 0 } ' " vdi do dai cO m2'". Do do, ngon ngii SAT kho c6 tht kigm chiing duoc trong thdi gian da thiic.
Nhiing khao sat nhu vay cho ta thay rang, v^n d i N P = co-NP
rat kho duqc giai quyet. Kho khan v l n la 6 cho, khi thttc hien nhiinf chiing minh, ta khong thg tranh diroc khau duyet toan bọ Ve hinh thiic, kho khan c6n duoc thg hien thong qua cac ket qua nghigr
ciiu mang tinh ly thuyet sau daỵ ^ Van de N P = co-NP lign quan den van de P = N P . Dudn ;
nhign, neu P = N P thi N P = co-NP, bdi v i P bat bien dudi tac dong ciia p"hep lay phan biị Nhung ngay ca khi P N P t h i k l i
nang N P = co-NP van c6 thg xay ra, va di nhign ca kha nang
N P 7^ co-NP ciing vaỵ
Tuy nhien, viec chiing minh N P 7^ c o - N P la khong he dc^n gian, bdi v i ta c6 dieu k h i n g dinh sau daỵ
D i n h ly 2.5.6 Neu N P co-NP M P ^ N P .
Chiing minh Bkng each chiing minh phan chiing va sii dung da0
thiic P = co-P ta CO ngay dieu can phai chiing minh. . ^
4
2.5 Cdu true cua cac Idp NP va co-NP
199 Digu do chiing to rang, gia thuyet N P ^ co-NP c6n manh hon ca gia thuygt P 7^ N P . Dau vSy, khong it chiing cii thuyet phuc ta t i n tucing rang hai gia thuyet nay nhieu kha nang se trd thanh hien thuc.
Mat khac, ve kha nang N P = co-NP, ta c6 ket luan nhu saụ
IDinh ly 2.5.7 Neu c6 mpt ngon ngii NP-đy du nao do ma phan bii cua no thuoc N P , thi N P = co-NP. bii cua no thuoc N P , thi N P = co-NP.
Chiing minh Gia sii LQ la ngon ngfl: NP-day du ma phan bii LQ^
thuoc N P .
Lay bat ky ngon ngii L trong N P . K h i do, do LQ la NP-day du, nen L LQ. Theo Dinh ly 2.5.5, ta c6 U". Do do, theo
gia thiet LQ" e N P va Dinh ly 2.4.4 (ii), / / e N P . Tit day suy ra
L e co-NP, nghia la N P C co-NP.
A Nguqc lai, doi v5i moi ngon ngii L G co-NP, ta c6 L'^ e N P . • Do Lo la NP-day du, ngn / / LQ. B5i vay, L LQ". Tvi gia thiet
• Lo" e N P suy ra G N P , nghia la co-NP C N P . •
W Dieu nguoc lai la hien nhign. Mot khi N P ^ co-NP t h i khong • chi mot ngon ngii NP-day du ma moi ngon ngii ciia N P deu c6