Алгоритмы для NP-трудных задач, осень 2009: Обзор

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  • NP- 1:

    .

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

    . (Computer Science ) 1. 1 / 24

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

  • 1 P NP

    2

    . (Computer Science ) 1. 2 / 24

  • 1 P NP

    2

    . (Computer Science ) 1. 2 / 24

  • , . .

    . (Computer Science ) 1. 3 / 24

  • P NP

    1 P NP

    2

    . (Computer Science ) 1. 4 / 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

    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- . . .

    . (Computer Science ) 1. 8 / 24

  • P NP

    NP- , , .

    NP- . . .

    . (Computer Science ) 1. 8 / 24

  • P NP

    NP- , , . NP- .

    . .

    . (Computer Science ) 1. 8 / 24

  • P NP

    NP- , , . NP- . .

    .

    . (Computer Science ) 1. 8 / 24

  • P NP

    NP- , , . NP- . . .

    . (Computer Science ) 1. 8 / 24

  • 1 P NP

    2

    . (Computer Science ) 1. 9 / 24

  • NP-

    , cn c.

    . (Computer Science ) 1. 10 / 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

  • , 1.7n , n0.

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

    , , .

    . (Computer Science ) 1. 11 / 24

  • 1 P NP

    2

    . (Computer Science ) 1. 12 / 24

  • (x y z) (x y) (y z) (z x) (x y z)

    . x , y , z , .

    . (Computer Science ) 1. 13 / 24

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

    . (Computer Science ) 1. 14 / 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.

    . (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.

    . (Computer Science ) 1. 15 / 24

  • n, n . , , 2n .

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

    . (Computer Science ) 1. 15 / 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

  • 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

    . (Computer Science ) 1. 18 / 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

    ,

    . (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

    ,

    . (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-.:

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