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
5.5.1Hieu suSt hau chac chan cua thuat toan
Trong muc nay ta khao sat hieu qua thuc su cua thuat toan tham lam cho bai toan NP-kho M A X - C L I Q U E .
Dg tien theo doi, ta nhac lai cac khai niem ve tinh hieu qua cua thuat toan xap x i . Cho IT la bai toan cue tigu hoa (hoac tuong ilng,
cue dai hoa) vdi tap d i i kien Dụ Gia sijf An la thuat toan xap x i
giai bai toan IT. Hieu suat RAn{d) cua thuat toan An tren dH kien
de Du duoc dinh nghia bdi dai luong sau day:
RAnid) = Au{d) /
OPTn{d) \ tuong ling,
OPTn{d) An{d)
Hieu suat tuyet doi RA„ cua thuat toan An, tilc hieu s u i t "trong
trudng hop xau nhat", duac xac dinh nhu sau:
RAU = inf { r > 1 : RAn{d) < r doi vdi moi d e Du}.
Tiep theo, dg thg hien hieu qua thuc su cua thuat toan, ta
dinh nghia khai niem hieu suit hdu chdc chdn {almost sure
performance ratio) R^^ cua An nhu sau:
R^^ = inf { r > 1 : RAn{d) < r doi vdi hau het moide Du}.
Quay t r d lai vdi bai toan M A X - C L I Q U E , bai toan t i m clique Idn nhat trong do t h i cho trudc, tiic t i m tap con C C 1/ vdi nhieu dinh
nhat trong do t h i G = {V, E) mh hai dinh bat ky ciia C deu ke
nhaụ Cac khao sat trong Phan 5.4 chi ra rang M A X - C L I Q U E la
mot bai toan rat kho, ngay ca khi giai gan dung. Moi thuat toan
xap xi giai M A X - C L I Q U E deu c6 hieu sudt tuyet doi b&ng oọ
Bay gid, ta xay dung mot thuat toan tham lam cho bai toan
M A X - C L I Q U E va chiing to rang thuat toan don gian nay c6 hi$u
sudt hdu chdc chdn < 2; nghia la trgn hau het mQi d i i kien cua bai
toan, thuat toan cho ta nghiem vdi gia t r i khong viigt qua khoang mSt niia gia t r i toi uụ
344 Cac giai phap Co vM thuat toan xap x i khac nhau cho bai toan M A X - C L I Q U E .
Sau day, theo phudng cham "gSp mon nao xao mon ay", t a xay dung
mot thuat toan t h a m lam kha đn gian, duoc ky hieu la Gruc, trong do M C la ten goi tat cua M A X - C L I Q U E . Thuat toan Gruc t i m ki6m
clique trong mot do t h i bat ky cho trudc,- b^ng each xuat phat t i t mot dinh duoc chgn t i i y y va tCtng budc b5 sung them mot dinh khac ke v6i t a t ca cac dinh da duoc chon trUdc dọ
Ro rang, tren m5i du: kien cua bai toan, tiic tren moi d5 t h i ,
thuat toan Gruc deu cho t a mot clique t o i dai (maximal). V i vay,
ta CO thg sijt dung cac ket qua khao sat ve lire luong cua cac clique (adsbygoogle = window.adsbygoogle || []).push({});
toi dai de danh gia hieu qua cua Gruc- Tuong t u n h u doi vdi hhi
toan tuong duong M A X - I S ve t i m kigm tap dOc lap (cac dinh khong kg nhau) Idn nhat, dudc khao sat trong [19], ta c6 thg ket luan r^ng
thuat toan Gruc c6 hieu suat hau chac ch^n RQIMC -
trong hau het m8i do t h i G, so dinh cua mot clique t o i dai bat ky nam trong khoang log n - 2 log log n va 2 log n - 2 log log n, d day
n la so dinh cua G.
Tuy nhien, nham m i n h hoa phuong phap neu tren cho viec danh gia t i n h hiiu hieu cua thuat toan noi chung va ciing dg tien theo doi, bay gid t a t r i n h bay tUOng d6i chi tiet each danh gia hieu suat cua thuat toan GTMC- Dau tien ta hay dien t a cu thg thuat toan tham lam naỵ
Gruc ~ " T r e n dau vao ( G ) , trong do G la do t h i v6 hudng vdi
tap dinh V: