# Алгоритмы для NP-трудных задач, осень 2009: Обзор

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• NP- 1:

.

Computer Science http://logic.pdmi.ras.ru/infclub/

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http://logic.pdmi.ras.ru/~infclub/

• 1 P NP

2

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• 1 P NP

2

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• , . .

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• P NP

1 P NP

2

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• P NP

P NP

(search problem) , I S , |I |. S (solution), (S , I ) = true.NP . , NP , .P , .P =NP? , , , ? Theoretical Computer Science.

. (Computer Science ) 1. 5 / 24

• P NP

P NP

(search problem) , I S , |I |. S (solution), (S , I ) = true.

NP . , NP , .P , .P =NP? , , , ? Theoretical Computer Science.

. (Computer Science ) 1. 5 / 24

• P NP

P NP

(search problem) , I S , |I |. S (solution), (S , I ) = true.NP . , NP , .

P , .P =NP? , , , ? Theoretical Computer Science.

. (Computer Science ) 1. 5 / 24

• P NP

P NP

(search problem) , I S , |I |. S (solution), (S , I ) = true.NP . , NP , .P , .

P =NP? , , , ? Theoretical Computer Science.

. (Computer Science ) 1. 5 / 24

• P NP

P NP

(search problem) , I S , |I |. S (solution), (S , I ) = true.NP . , NP , .P , .P =NP? , , , ? Theoretical Computer Science.

. (Computer Science ) 1. 5 / 24

• P NP

, A (reduces to) B , A B , B A.-: A A B , B . B A. NP- (NP-complete), . NP. ( ), .

. (Computer Science ) 1. 6 / 24

• P NP

, A (reduces to) B , A B , B A.

-: A A B , B . B A. NP- (NP-complete), . NP. ( ), .

. (Computer Science ) 1. 6 / 24

• P NP

, A (reduces to) B , A B , B A.-: A A B , B . B A.

NP- (NP-complete), . NP. ( ), .

. (Computer Science ) 1. 6 / 24

• P NP

, A (reduces to) B , A B , B A.-: A A B , B . B A. NP- (NP-complete), . NP.

( ), .

. (Computer Science ) 1. 6 / 24

• P NP

, A (reduces to) B , A B , B A.-: A A B , B . B A. NP- (NP-complete), . NP. ( ), .

. (Computer Science ) 1. 6 / 24

• P NP

P vs NP

, P=NP., , :http://www.win.tue.nl/gwoegi/P-versus-NP.htm P =NP NP- .

. (Computer Science ) 1. 7 / 24

http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

• P NP

P vs NP

, P=NP.

, , :http://www.win.tue.nl/gwoegi/P-versus-NP.htm P =NP NP- .

. (Computer Science ) 1. 7 / 24

http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

• P NP

P vs NP

, P=NP., , :http://www.win.tue.nl/gwoegi/P-versus-NP.htm

P =NP NP- .

. (Computer Science ) 1. 7 / 24

http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

• P NP

P vs NP

, P=NP., , :http://www.win.tue.nl/gwoegi/P-versus-NP.htm P =NP NP- .

. (Computer Science ) 1. 7 / 24

http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

• P NP

NP- , , . NP- . . .

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• P NP

NP- , , .

NP- . . .

. (Computer Science ) 1. 8 / 24

• P NP

NP- , , . NP- .

. .

. (Computer Science ) 1. 8 / 24

• P NP

NP- , , . NP- . .

.

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• P NP

NP- , , . NP- . . .

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• 1 P NP

2

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• NP-

, cn c.

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• , 1.7n , n0.

The hardware approach: 10 . n0 + 4.The brainware approach: 1.3n. 2 n0.

, , .

. (Computer Science ) 1. 11 / 24

• , 1.7n , n0.

The hardware approach: 10 . n0 + 4.

The brainware approach: 1.3n. 2 n0.

, , .

. (Computer Science ) 1. 11 / 24

• , 1.7n , n0.

The hardware approach: 10 . n0 + 4.The brainware approach: 1.3n. 2 n0.

, , .

. (Computer Science ) 1. 11 / 24

• , 1.7n , n0.

The hardware approach: 10 . n0 + 4.The brainware approach: 1.3n. 2 n0.

, , .

. (Computer Science ) 1. 11 / 24

• 1 P NP

2

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• (x y z) (x y) (y z) (z x) (x y z)

. x , y , z , .

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• (Boolean satisfiabilityproblem, SAT): , , , . (satisfying assignment), , , (satisfiable).

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• n, n . , , 2n .

, . t {0, 1}n d (t, d) ( t d) , d t.

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• n, n . , , 2n .

, . t {0, 1}n d (t, d) ( t d) , d t.

. (Computer Science ) 1. 15 / 24

• n, n . , , 2n .

, . t {0, 1}n d (t, d) ( t d) , d t.

. (Computer Science ) 1. 15 / 24

• n, n . , , 2n .

, .

t {0, 1}n d (t, d) ( t d) , d t.

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• n, n . , , 2n .

, . t {0, 1}n d (t, d) ( t d) , d t.

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• F 3- (t, d) 3d :

t F , C = (x1 x2 x3) , t x1, x2 x3

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• F 3- (t, d) 3d :

t F , C = (x1 x2 x3)

, t x1, x2 x3

. (Computer Science ) 1. 16 / 24

• F 3- (t, d) 3d :

t F , C = (x1 x2 x3) , t x1, x2 x3

. (Computer Science ) 1. 16 / 24

• F 3- (t, d) 3d :

t F , C = (x1 x2 x3) , t x1, x2 x3

. (Computer Science ) 1. 16 / 24

• 3n-

Simple-3-SAT(F )

, (0n, n/2),(1n, n/2)

Simple-3-SAT 3n, n = n(F )

F .

, , .

. (Computer Science ) 1. 17 / 24

• 3n-

Simple-3-SAT(F ), (0n, n/2),(1n, n/2)

Simple-3-SAT 3n, n = n(F )

F .

, , .

. (Computer Science ) 1. 17 / 24

• 3n-

Simple-3-SAT(F ), (0n, n/2),(1n, n/2)

Simple-3-SAT 3n, n = n(F )

F .

, , .

. (Computer Science ) 1. 17 / 24

• 3n-

Simple-3-SAT(F ), (0n, n/2),(1n, n/2)

Simple-3-SAT 3n, n = n(F )

F .

, , .

. (Computer Science ) 1. 17 / 24

• 1 P NP

2

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• 3-

3- (3-clique problem) 3- ( ) .

Matrix-Clique(G ) A G A3

,

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• 3-

3- (3-clique problem) 3- ( ) .

Matrix-Clique(G )

A G A3

,

. (Computer Science ) 1. 19 / 24

• 3-

3- (3-clique problem) 3- ( ) .

Matrix-Clique(G ) A G

A3

,

. (Computer Science ) 1. 19 / 24

• 3-

3- (3-clique problem) 3- ( ) .

Matrix-Clique(G ) A G A3

,

. (Computer Science ) 1. 19 / 24

• 3-

3- (3-clique problem) 3- ( ) .

Matrix-Clique(G ) A G A3

,

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• 3-

3- (3-clique problem) 3- ( ) .

Matrix-Clique(G ) A G A3

,

. (Computer Science ) 1. 19 / 24

• O(n), 2.376 (matrix multiplication exponent).

: Ak [i , j ] k i j G ( )3- 3 A3

. (Computer Science ) 1. 20 / 24

• O(n), 2.376 (matrix multiplication exponent).

: Ak [i , j ] k i j G ( )3- 3 A3

. (Computer Science ) 1. 20 / 24

• O(n), 2.376 (matrix multiplication exponent).

: Ak [i , j ] k i j G ( )

3- 3 A3

. (Computer Science ) 1. 20 / 24

• O(n), 2.376 (matrix multiplication exponent).

: Ak [i , j ] k i j G ( )3- 3

A3

. (Computer Science ) 1. 20 / 24

• O(n), 2.376 (matrix multiplication exponent).

: Ak [i , j ] k i j G ( )3- 3 A3

. (Computer Science ) 1. 20 / 24

• (maximum cut proble MAX-CUT) , , , .

3- G n .

H 3 2n/3 : G w , H 3- w . 3-.: