Эффективные алгоритмы, осень 2007: Подходы к решению NP-трудных задач

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  • / 13: NP-

    .

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

    . (CS ) 13. NP- 1 / 58

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

  • 1

    ( )

    2

    3 3-k-

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    6 3-

    7 3-

    . (CS ) 13. NP- 2 / 58

  • 1

    ( )2

    3 3-k-

    4

    5

    6 3-

    7 3-

    . (CS ) 13. NP- 2 / 58

  • 1

    ( )2

    3

    3-k-

    4

    5

    6 3-

    7 3-

    . (CS ) 13. NP- 2 / 58

  • 1

    ( )2

    3

    3-k-

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    6 3-

    7 3-

    . (CS ) 13. NP- 2 / 58

  • 1

    ( )2

    3

    3-k-

    4

    5

    6 3-

    7 3-

    . (CS ) 13. NP- 2 / 58

  • 1

    ( )2

    3

    3-k-

    4

    5

    6 3-

    7 3-

    . (CS ) 13. NP- 2 / 58

  • 1

    ( )2

    3

    3-k-

    4

    5

    6 3-

    7 3-

    . (CS ) 13. NP- 2 / 58

  • , 1.7n , n0.

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

    , , .

    . (CS ) 13. NP- 3 / 58

  • , 1.7n , n0.

    The hardware approach: 10 . n0 + 4.

    The brainware approach: 1.3n. 2 n0.

    , , .

    . (CS ) 13. NP- 3 / 58

  • , 1.7n , n0.

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

    , , .

    . (CS ) 13. NP- 3 / 58

  • , 1.7n , n0.

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

    , , .

    . (CS ) 13. NP- 3 / 58

  • NP- , , . NP- . . .

    . (CS ) 13. NP- 4 / 58

  • NP- , , .

    NP- . . .

    . (CS ) 13. NP- 4 / 58

  • NP- , , . NP- .

    . .

    . (CS ) 13. NP- 4 / 58

  • NP- , , . NP- . .

    .

    . (CS ) 13. NP- 4 / 58

  • NP- , , . NP- . . .

    . (CS ) 13. NP- 4 / 58

  • 1

    ( )2

    3

    3-k-

    4

    5

    6 3-

    7 3-

    . (CS ) 13. NP- 5 / 58

  • , ( ) .

    . (CS ) 13. NP- 6 / 58

  • 1

    ( )2

    3

    3-k-

    4

    5

    6 3-

    7 3-

    . (CS ) 13. NP- 7 / 58

  • , , , ,

    . (CS ) 13. NP- 8 / 58

  • ,

    , , ,

    . (CS ) 13. NP- 8 / 58

  • , , , ,

    . (CS ) 13. NP- 8 / 58

  • , , , ,

    . (CS ) 13. NP- 8 / 58

  • , , , ,

    . (CS ) 13. NP- 8 / 58

  • NP-

    NP- ,

    , , .

    . (CS ) 13. NP- 9 / 58

  • NP-

    NP- ,

    , , .

    . (CS ) 13. NP- 9 / 58

  • NP-

    NP- ,

    , , .

    . (CS ) 13. NP- 9 / 58

  • NP-

    NP- ,

    , , .

    . (CS ) 13. NP- 9 / 58

  • NP-

    NP- ,

    , , .

    . (CS ) 13. NP- 9 / 58

  • , , , 1.619K , K .

    . (CS ) 13. NP- 10 / 58

  • , (1, 2)- ( , T (K ) T (K 1) + T (K 2) + poly(K )) ( ) , , , :

    (x a . . . ) (x b . . . ) (x c . . . ) . . .XXXXXXXXz

    9x = 1 x = 0

    (c . . . ) . . . (a . . . ) (b . . . ) . . .

    . (CS ) 13. NP- 11 / 58

  • , (1, 2)- ( , T (K ) T (K 1) + T (K 2) + poly(K ))

    ( ) , , , :

    (x a . . . ) (x b . . . ) (x c . . . ) . . .XXXXXXXXz

    9x = 1 x = 0

    (c . . . ) . . . (a . . . ) (b . . . ) . . .

    . (CS ) 13. NP- 11 / 58

  • , (1, 2)- ( , T (K ) T (K 1) + T (K 2) + poly(K )) ( )

    , , , :

    (x a . . . ) (x b . . . ) (x c . . . ) . . .XXXXXXXXz

    9x = 1 x = 0

    (c . . . ) . . . (a . . . ) (b . . . ) . . .

    . (CS ) 13. NP- 11 / 58

  • , (1, 2)- ( , T (K ) T (K 1) + T (K 2) + poly(K )) ( ) , , , :

    (x a . . . ) (x b . . . ) (x c . . . ) . . .XXXXXXXXz

    9x = 1 x = 0

    (c . . . ) . . . (a . . . ) (b . . . ) . . .

    . (CS ) 13. NP- 11 / 58

  • , , (1, 1)-, .

    . (CS ) 13. NP- 12 / 58

  • ?

    (x . . .) (x . . .) . . .)

    (x) (x . . .) . . .

    PPPPq

    y (x . . . )

    (x y . . . ) (x y . . . ) . . .

    (x y . . . ) (x . . . ) (y . . . ) . . .

    (2, 2)- x

    (2, 1)- x

    . (CS ) 13. NP- 13 / 58

  • ?

    (x . . .) (x . . .) . . .

    )

    (x) (x . . .) . . .

    PPPPq

    y (x . . . )

    (x y . . . ) (x y . . . ) . . .

    (x y . . . ) (x . . . ) (y . . . ) . . .

    (2, 2)- x

    (2, 1)- x

    . (CS ) 13. NP- 13 / 58

  • ?

    (x . . .) (x . . .) . . .)

    (x) (x . . .) . . .

    PPPPq

    y (x . . . )

    (x y . . . ) (x y . . . ) . . .

    (x y . . . ) (x . . . ) (y . . . ) . . .

    (2, 2)- x

    (2, 1)- x

    . (CS ) 13. NP- 13 / 58

  • ?

    (x . . .) (x . . .) . . .)

    (x) (x . . .) . . .

    PPPPq

    y (x . . . )

    (x y . . . ) (x y . . . ) . . .

    (x y . . . ) (x . . . ) (y . . . ) . . .

    (2, 2)- x

    (2, 1)- x

    . (CS ) 13. NP- 13 / 58

  • ?

    (x . . .) (x . . .) . . .)

    (x) (x . . .) . . .

    PPPPq

    y (x . . . )

    (x y . . . ) (x y . . . ) . . .

    (x y . . . ) (x . . . ) (y . . . ) . . .

    (2, 2)- x

    (2, 1)- x

    . (CS ) 13. NP- 13 / 58

  • ?

    (x . . .) (x . . .) . . .)

    (x) (x . . .) . . .

    PPPPq

    y (x . . . )

    (x y . . . ) (x y . . . ) . . .

    (x y . . . ) (x . . . ) (y . . . ) . . .

    (2, 2)- x

    (2, 1)- x

    . (CS ) 13. NP- 13 / 58

  • ?

    (x . . .) (x . . .) . . .)

    (x) (x . . .) . . .

    PPPPq

    y (x . . . )

    (x y . . . ) (x y . . . ) . . .

    (x y . . . ) (x . . . ) (y . . . ) . . .

    (2, 2)- x

    (2, 1)- x

    . (CS ) 13. NP- 13 / 58

  • ?

    (x . . .) (x . . .) . . .)

    (x) (x . . .) . . .

    PPPPq

    y (x . . . )

    (x y . . . ) (x y . . . ) . . .

    (x y . . . ) (x . . . ) (y . . . ) . . .

    (2, 2)- x

    (2, 1)- x

    . (CS ) 13. NP- 13 / 58

  • ?

    (x . . .) (x . . .) . . .)

    (x) (x . . .) . . .

    PPPPq

    y (x . . . )

    (x y . . . ) (x y . . . ) . . .

    (x y . . . ) (x . . . ) (y . . . ) . . .

    (2, 2)- x

    (2, 1)- x

    . (CS ) 13. NP- 13 / 58

  • 1

    ( )2

    3

    3-k-

    4

    5

    6 3-

    7 3-

    . (CS ) 13. NP- 14 / 58

  • - , , .

    . (CS ) 13. NP- 15 / 58

  • 1

    ( )2

    3

    3-k-

    4

    5

    6 3-

    7 3-

    . (CS ) 13. NP- 16 / 58

  • PPSZ-

    , , ,

    . (CS ) 13. NP- 17 / 58

  • PPSZ-

    , , ,

    . (CS ) 13. NP- 17 / 58

  • PPSZ-

    , ,

    ,

    . (CS ) 13. NP- 17 / 58

  • PPSZ-

    , , ,

    . (CS ) 13. NP- 17 / 58

  • PPSZ-

    PPSZ-SAT(F )

    {1, . . . , n} 2n/3,

    I ,I , ,

    - , ,

    . (CS ) 13. NP- 18 / 58

  • PPSZ-

    PPSZ-SAT(F )

    {1, . . . , n}

    2n/3,

    I ,I , ,

    - , ,

    . (CS ) 13. NP- 18 / 58

  • PPSZ-

    PPSZ-SAT(F )

    {1, . . . , n} 2n/3,

    I ,I , ,

    - , ,

    . (CS ) 13. NP- 18 / 58

  • PPSZ-

    PPSZ-SAT(F )

    {1, . . . , n} 2n/3,

    I ,

    I , ,

    - , ,

    . (CS ) 13. NP- 18 / 58

  • PPSZ-

    PPSZ-SAT(F )

    {1, . . . , n} 2n/3,

    I ,I , ,

    - , ,

    . (CS ) 13. NP- 18 / 58

  • PPSZ-

    PPSZ-SAT(F )

    {1, . . . , n} 2n/3,

    I ,I , ,

    - , ,

    . (CS ) 13. NP- 18 / 58

  • PPSZ-

    PPSZ-SAT(F )

    {1, . . . , n} 2n/3,

    I ,I , ,

    - , ,

    . (CS ) 13. NP- 18 / 58

  • 1

    ( )2

    3

    3-k-

    4

    5

    6 3-

    7 3-

    . (CS