? tic-tac-toe???

1
?????????????????????????????

• ???? ???????
• ??? ??,?? ??

2
?????
? tic-tac-toe??? ? ??????????? ? HEX??? ?
????????????? ? PATH-WITH-FP???NP??? ? ?????
3
tic-tac-toe???
tic-tac-toe ???
????2????????
????????





?,?,??????????
?????????????????
4
tic-tac-toe???????????
N ?4 ??? ????????????????
????? ??????????? ??????????
11 1 8 1 12
6 2 2 9 10
3 7 9 3
6 7 4 4 10
12 5 8 5 11
??????????2????? ????????????
???????
????????????????????
?????????????? ??????????????????????
5
?????
???????????????????(forbidden pair)
???????????????? ???????
11 1 8 1 12
6 2 2 9 10
3 7 9 3
6 7 4 4 10
12 5 8 5 11
?????(perfect forbidden pairs)
6
????????
???????P ,?????(??)????V ???
P (p1, q1) ,???, (pk , qk) (1?k ? )
pi , qi ?V (1?i?k)
???,??????disjoint??????
??winning path??? P ??????? pi , qi???
P ?????????
7
HEX???
red side
blue side
red player ? blue player ????????????????? ???side
?????player???
red side
blue side
8
HEX???
red side
blue side
red side
blue side
9
HEX???
???????
?????????winning strategy???
??????,winning strategy???player???????
PSPACE-complete
10
HEX???????????
???
0
1
0
6
2
1
6
7
3
2
7
11
8
4
3
8
11
12
9
5
4
9
12
15
13
5
10
10
13
15
16
14
14
16
18
17
17
18
19
19
20
20
N ??
(N-1)??
11
HEX???????????
???
red player? blue player??????? ?????????????
0
1
0
6
2
1
??????!
6
7
3
2
7
11
8
4
3
8
11
12
9
5
4
blue player ??????? ????????????
9
12
15
13
5
10
10
13
15
16
14
14
16
18
17
17
18
19
19
20
red player?????!
20
?????1???????????????????
12
HEX?????????
HEX??????????
???????????????????????
13
??????????????
shannon vertex switching game
2????????????????G (V ,E )??????
????? s ? t ?????? ???????????????
G
t
s
14
??????????????
shannon vertex switching game
??????????winning strategy??????????
PSPACE-complete
G
t
s
15
PERFECT-FP??
????????
???????????
PERFECT-FP (G, s, t ) G has perfect
forbidden pairs
??? G (V, E) ,???s,???t ???
????????
PATH-WITH-FP (G, s, t, P ) G has a path from
s to t that contains at most one vertex
from each pair in P
??? P (p1, q1) ,???, (pk , qk) (1?k ?
) ,
????????????? perfect forbidden pairs ???????
pi , qi ?V (1?i?k) ???
16
PERFECT-FP??
??(G, s, t )
PERFECT-FP
accept
reject
17
PATH-WITH-FP?NP???
??
PATH-WITH-FP ?NP????? HAROLD?,1976
1.?PATH-WITH-FP?NP? ???
PATH-WITH-FP???????
?????

• ? ?????s-t???????
• ? ????????????????????????
• ????????,??????????

????????????????
PATH-WITH-FP?NP ???
18
PATH-WITH-FP?NP???
??
PATH-WITH-FP ?NP????? HAROLD?,1976
2.?PATH-WITH-FP ?NP-hard ? ???
3SAT ?? PATH-WITH-FP ???
19
PATH-WITH-FP?NP???
??
s
t
??? ?????m?? ??? ?????n???????
CNF?? ?n?, ?m???
20
PATH-WITH-FP?NP???
s
t
??? ?????m?? ??? ?????n???????
CNF?? ?n?, ?m???
21
PATH-WITH-FP?NP???
s
t
?? x ? ???????
??? x ? ????????????????????????
22
PATH-WITH-FP?NP???
s
t
PATH-WITH-FP ?????
? x1 1, x2 0, x3 1, x4 0 ???,f 1
PATH-WITH-FP ?NP??(???)
23
PERFECT-FP??
??(G, s, t )
PERFECT-FP
PERFECT-FP?NPNP
NP????
accept
reject
24
?????

• PERFECT-FP????????
• ????????????????????????????

?. series-parallel graph