31' : :

1
?????? 31. ???? ?????????????? ??????
??????????? ?????? ???. ???????????????? ????
? ???????, ????????? ???????????? ?
??????????? ?????????? ??????-???????????
???????? (???????????)
???? ?. ????? ???????? ??????? ????????
??????????, ???
Email mslevin_at_acm.org / mslevin_at_iitp.ru
PLAN 1.?????? ???????????? ???????????? ?
??????????? , ???????????, ?????????? 2.??????
? ?????? ????????????? ???????????
(????????????, 3-????????????, ?????????
????????, 3-?????????????(matching), ?????,
??????????? ????, ?????????) 3.??????
??????????? ??????????? ?????? (Timetabling
problems) ???????????? ? ???????????,
??????????, ????????? ? ????? ???????,
???????????? (??????, ?????)
???. 10, 2004
2
?????? ????????????
?????? ?????????? U x1, x2, x3, ( 0 or
1 )
?????????? ???????? (AND, OR, NOT)
????????? (???) NOT ??????????? (?)conjunction
AND, ?????? ??? ???????????
????????? ?????????? ????????? C ??????? C1
( NOT x1) OR x2 C2
x1 OR (NOT x3) C3 (NOT
x5) OR (NOT x7) OR x9 C4
x1 OR x10 OR (NOT x11) OR x22
??? TRUE (1) (?????)
3
?????? ????????????
?????? ???????????? ???????? ??????? NP-??????
????????? (???????) ????????? ? ?????????
???????? ?????????? ???????????, ??????,
?????????????? ?????????????? ?????? ?????
????, ??? ?????? ????? ? ????? ?????? ?????? ???
???????? ???????? ??? ??????????????
NP-??????? ??? ?????? ?????
4
?????? ????????????
?????????? 1.??????????? ????????
(Software Verification) 2?????????????? ?
??????????? ? ??????????? (Electronic Design
Automation and Verification) 3.?????? ???????
(Model Analysis) 4.???????? ??????? (Model
Checking) 5.?????????????? ?????? (Theorem
Prover ) 6.???????????? ? ?????????????
?????????? (AI Planning)
5
?????? ???????????? ??????????? ??? ??????????
? ???????????????? ? ???????????
?????? ?????????? ?? xo(x1,,xn) ????? ???
y(xo) 1 ??? ???
???????
??????? xi / not xi
x1
c1
x2
c2
y (0 or 1)
. . .
y c1c2 cn
cn-1
xm-1
cn
xm
?????? c1 not x1 OR x2 OR x4 OR not
x5 OR x7 c2
x1 OR not x2 OR not x3 OR x5 OR x7
c3 not x1 OR not x2 OR x3 OR
not x5 OR not xn c4
not x2 OR x3 OR x7 OR xn-2 OR xn-1
.
. .
6
?????? 3-????????????
???????
?????? 3-???????????? ? ?????? cj - 3 ????????
??????? xi / not xi
x1
c1
x2
c2
y (0 or 1)
. . .
cn-1
y c1c2 cn
xm-1
cn
xm
?????? c1 not x1 OR x2 OR x4
c2 x2 OR not x3 OR
x7 c3 not x1 OR
not x5 OR not xn c4
not x2 OR xn-2 OR xn-1
. . .
7
????????? ??????? ????????????? ?????
6 ??????? NP-?????? ?????
????????????
3-????????????
3-?????????????
????????? ????????
????????? (???????? ??????)
???? ??????????
?????
8
3-?????????????
A B C
A a1, an
B b1, bn
C c1, cn
?????? ?????????? ?? ???????? ?????? ????????
(??? ???????????) ??? ???
9
?????????
????????? ????????? A 1, , i , , n
???? ????????? b1, , bi , , bn
???????? ?????????? ?? A ? A ????? ???
? bi ? bj
i?A j?A\A
??? ???
10
???????? ????????
????????? ?????? A a1, an , ?????????
????? Ee1, ,ek, ???? G (A, E)
?????? ????? ????????? ???????? (A ? A)
??????? ????????? ?e ? E
11
???? ??????????
?????? ?????????? ?? ???? ?????????? ??? ???
12
?????
k 4
?????? ?????????? ?? ???????-????? ? ??????
?????? k ??? ???
13
??????????? ????? ????? ????????????
???????????? (Timetabling)
??????? ( ???????) ????????????? ??????
1.????????? ????? 2.??????????/ ?????????? /
????????????? 3.?????? ? ????????? 4.??????
?????? ?????????? 5.??????????? ?????? ??????
?????????? 6.????? 7.???????????? ????????????
(Combinatorial design, ????????, ?????????
???????? - Latin square ? ??.) ??.
14
??????????? ????? ????? ????????????
???????????? (Timetabling)
????????? ? ??????????????? ????? 1.??????????
?????? 2.?????? ?????????????? ?????????
?????????? ??????? (?? ?????? ?????? ?????????
? ??.) ????????? ?? ?????? ??? ??????
????????? ???? 3.?????? ?? ?????? ????????
??????????? (CONSTRAINED PROGRAMMING) 4.????-????
????? (????????? ?????? ????????? ???????????)
??????????? ?? ?????? ????????????? ???????? (
ant colony optimization,) ???????????
????????? ????? (iterated local search) ?????
?????? Tabu ????? ???????????? ?????????
5.???????????? ???????????? (COMBINATORIAL
DESIGN) 6.???????????? ?????????? (???????
multi-objective evolutionary optimization) 7.?????
???? ????? ???????
15
?????? ?????????? (?????? 30)
?????????? (??????????, ?????????????,
????????????????)
a
???????????
???????? (????????, ????????????)
b
matrix of weights cij
1
a b c d e f g h
2
c
???????
. . . . . . . . .
. . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . .
. . . . . . . . .
. . . . . . . .
. . . .
1 2 3 4 5 6 7 8
3
d
4
e
5
f
6
7
g
8
h
??????????? ????
16
?????? ?????????? (?????? 30)
?????????? (??????????, ?????????????,
????????????????)
a
???????????
???????
b
matrix of weights cij
1
a b c d e f g h
2
c
?????
. . . . . . . . .
. . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . .
. . . . . . . . .
. . . . . . . .
. . . .
1 2 3 4 5 6 7 8
3
d
4
e
5
f
6
7
g
8
h
??????????? ????
17
?????? ?????????? (?????? 30)
?????????? (??????????, ?????????????,
????????????????)
a
???????????
?????? ?????????
b
matrix of weights cij
1
a b c d e f g h
?????????
2
c
. . . . . . . . .
. . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . .
. . . . . . . . .
. . . . . . . .
. . . .
1 2 3 4 5 6 7 8
3
d
4
e
5
f
6
7
g
8
h
??????????? ????
18
?????? ? ?????????? AP (?????? 30)
b1
a1
b2
a2
b3
????????
???????
a3
. . .
. . .
an
bm

• ???????????? (??????????????)
• ???????? (????????, ????????????, ???????) A
  a1 , , ai , , an
• ??????? (???????, ??????????) B b1 , , bj
  , . bm
• (????? n m)
• ????????????? ???? ai ? bj is c ( ai , bj )
• xij 1 i???? ai ???????? ?? ??????? bj ?
  0 ? ????????? ?????? ( xij ? 0,1 )
• ?????? max ?ni1 ?nj1 cij xij
• s.t. ?ni1 xij
  1 ? j
• 
  ?nj1 xij 1 ? i

19
???????????? ?????? ? ?????????? (Quadratic
Assignment Problem QAP) (?????? 30)
b1
a1
b2
a2
b3
????????
???????
a3
. . .
. . .
an
bm
??????? ????? (??????) cij
??????? ?????????? dij
b1 bj bn
b1 bj bn
. . . .
. . . cij
. . .
. . . .
a1 ai an
. . . .
. . . dij
. . .
. . . .
b1 bi bn
20
???????????? ?????? ? ?????????? (Quadratic
Assignment Problem QAP)
??????? (?????????) ??????????????
???????????? ?????????? ????????? i
i?1,,n ?? ??????? p(i) , ??? p
???????????? ??????? 1,,n , ?????????
????????? ???????????? ? p .
?????????? ??? n ?? n ??????? (i)???????
?????? (??? ?????????? ) C ??? ???????
(i,j) ???????????? ????? ????? ??????????
(????????, ????????????? ) i ? j,
(ii)??????? ?????????? D ??? (i,j)
(??????? p(i), p(j) ) ???????????? ??????????
????? ????????? p(i) and p(j). ?????
???????, ?????? QAP ????? ???
max ?ni1 ?nj1 cij dp(i)p(j) p ? ?
21
???????????? ?????? ? ?????????? (Quadratic
Assignment Problem QAP)
??????? (?????????)
??????? (?????, ????????????)
a
????????s (?????? ? ??.)
??????????? (????-?????)
b
1
1
2
c
2
3
d
3
4
e
4
5
5
6
f
6
7
g
8
h
????-??????? ????
22
?????????? ?????? ? ?????????? GAP (?????? 30)
i
j
k as j (1m)
????-?????
b1
a1
b2
a2
b3
???????? (???????)
??????? (??????, ?????????? ? ??. )
??????? Rk
a3
. . .
. . .
an
bm

• ???????????? (??????????????)
• ???????? (????????, ????????????, ???????) A
  a1 , , ai , , an
• ??????? (?????????? ? ??.) B b1 , , bj ,
  . bm
• (????? n m)
• ????????????? ???? ai ? bj is c ( ai , bj )
• xij 1 ???? ai ????????? ?? ??????? bj ?
  0 ? ????????? ?????? ( xij ? 0,1 )
• ?????? 1 max ?ni1 ?nj1 cij xij
• s.t. ?ni1 rik
  xij ? Rj ? j ( Rk ??? ?????? ??????
  k)
• 
  ?nj1 xij 1 ? i

23
?????????? ?????? ? ?????????? GAP (?????
???????) (?????? 30)
i
j
k (1K)
one-many
b1
a1
b2
a2
b3
???????? (???????)
??????? (????????, ??????????)
??????? Rjk
a3
. . .
. . .
an
bm

• ???????????? (??????????????)
• ???????? (e.g., personnel, facilities, tasks) A
  a1 , , ai , , an
• ??????? (?????????? ? ??.) B b1 , , bj ,
  . bm
• (????? n m)
• ????????????? ???? ai ? bj c ( ai , bj )
• xij 1 ???? ai ???????? ?? ??????? bj ?
  0 ? ????????? ?????? ( xij ? 0,1 )
• ?????? 2 max ?ni1 ?nj1 cij xij
• s.t. ?mj1 ?ni1 rik xij
  ? Rk ? k (K ????? ???????)
• 
  ?nj1 xij 1 ? i

24
??????????? ????? ????? ????????????
???????????? (Timetabling) ???? ????? (?????)
A a1, an
B b1, bm
?????????????
????? (??? ?????? ?????????)
?????? 3-????????????? (3-? ??????? ????)
????????? (?????)
C c1, ck
0
t
T
2T
25
??????????? ????? ????? ????????????
???????????? (Timetabling) ???????????
??????????? (???????) ??? ???????? ?????? 2
????? ??????????? ?????? ? ??????????? ? ???????
(lecturers - time) ?????? 11 ?????? ???? ?????
??????? 12 (?? ???????) Lecturers 5 and 7 can
teach only in auditoriums 9 or 10
(lecturers-auditoriums) ??? ?????
????????? ?????? 1 ????????? ????????? 5 ?
??????????? ????? (?????? - ?????) ?????? 7, 8,
9 ?????? ????? ???? ?? ??????? 1 (???? ?? ????)
(?????? - ???????)
(?????? - ??????) ?????? 4 ???????????? ???????
10 (?????? - ???????) ?????? 5 ??????? ????????
5 8 ??? ??????? ?? ??????? (?????? -
???????) ??? ????????? ????????? 1 ??????? ?
????? (????????? - ?????) ????????? 4
????????????? ?????? ??????? 5, 7, 8, 9
(????????? - ??????)
26
??????????? ????? ????? ????????????
???????????? (Timetabling) ???????????
???? ??????????? (?? ?????????) ???????
??????? ??????? - ?????? ??????? -
????????? ??????? - ????? ?????? - ?????? ??????
- ????????? ?????? - ????? ?????????
????????? ????????? - ????? ???? ??????????? (??
?????) ?????????? (???????? ????????? ?
??.) ?????????????? (??????????? ?? ???????? ?
??.)
27
??????????? ????? ????? ????????????
???????????? (Timetabling) ???????????
A a1, an
B b1, bm
????? (?????? ?????????)
?????????????
????????? (?????)
C c1, ck
0
t
T
2T
28
????????????? ?????????? ??? timetabling
1.?????? ?????? ?????????? ? ????????????????
???????? (COMMUNICATION
TIMETABLING) 2.?????? ?????? ?????????? ?
?????????????? ???????? (MONITORING
TIMETBLING)
29
Timetabling ?????? ? ?????? (?????????)
G.L. Nemhauser, M.A. Trick, Scheduling a major
college basketball conference. Operations
Research, 46(1), 1-8, 1998.
????????????, ?????? ???????????
1.??????? ???????
???????, slots (weekday slots - ???? ? ????????
??????, ???? ? ????? ?????? -
weekend slots), ?????? ??????? ?????? ?????? ?
??????, 8 home slots ?????? ????, 8 away slots
- ?????? ?? ?????? 2. ??????????? ??? ?????? ?
?????? home slots, away slots, ???????
home-away-? ?.?. 3. Patterns ?
??????????? lt 2 away games (???), lt 2 home
games (???) 4.??????????? ??? ???
?????? ???? ?????????? (?? ???????????
??????????) ????? ???????

??? 1. A pattern
??? ?????? H (home), A (away), B (bye).
??????? HAA, HBW
????? ????????? pattern, ?????? HHA, AHA, HAH,
AAH (??????? a, b, c, d) (enumeration integer
programming) ??? 2. ????????? ???? ??
patterns. ??? ???? timetabling
(integer programming)
??? 3. ????????? ??????? ??
patterns. ??? ???? ??????????? ????
(quadratic assignment problem)
?????????? 24 ?????