Приближенное решение задач комбинаторной оптимизации: алгоритмы и трудность, осень 2016: Теорема Хостада

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  • :

    9:

    .

    . ..

    -, Computer Science Club, 2016

    . ( ) 9: , CSclub, 2016 1 / 26

  • :

    MAX-3LIN: F2,

    .: ,

    .

    0 1/2. MAX-3LIN 1/2.

    > 0 MAX-3LIN(1 , 1/2 + ) NP-.

    . ( ) 9: , CSclub, 2016 2 / 26

  • :

    MAX-3LIN: F2,

    .: ,

    .

    0 1/2. MAX-3LIN 1/2.

    > 0 MAX-3LIN(1 , 1/2 + ) NP-.

    . ( ) 9: , CSclub, 2016 2 / 26

  • :

    MAX-3LIN: F2,

    .: ,

    .

    0 1/2. MAX-3LIN 1/2.

    > 0 MAX-3LIN(1 , 1/2 + ) NP-.

    . ( ) 9: , CSclub, 2016 2 / 26

  • :

    MAX-3LIN: F2,

    .: ,

    .

    0 1/2. MAX-3LIN 1/2.

    > 0 MAX-3LIN(1 , 1/2 + ) NP-.

    . ( ) 9: , CSclub, 2016 2 / 26

    MAX-3LIN(1, ) P.

  • :

    : 3 1 x , y Fn2 .2 z N(y), = 1 2.3 f x , y , x + z .4 , f (x) + f (y) = f (x + z).

    .

    . MAX-LCk(1, ) 6p MAX-3LIN(1 , 1/2 + ).

    . ( ) 9: , CSclub, 2016 3 / 26

  • :

    : 3 1 x , y Fn2 .2 z N(y), = 1 2.3 f x , y , x + z .4 , f (x) + f (y) = f (x + z).

    .

    . MAX-LCk(1, ) 6p MAX-3LIN(1 , 1/2 + ).

    . ( ) 9: , CSclub, 2016 3 / 26

  • PCP MAX-LCk(1, ):

    G (V ,E , [k], f ) . (v) [k] . Xi : (x1, . . . , xk) 7 xi :

    Xi (x1, . . . , xk) = Xi (x1, . . . ,xk).

    , . :

    1 + Xi (x1, . . . , xk) = Xi (1 + x1, . . . , 1 + xk)

    2.

    PCP Tv : {0, 1} {0, 1}, v V .

    . ( ) 9: , CSclub, 2016 4 / 26

  • PCP A MAX-LCk(1, )

    1 (u, v) E (G ).2 Tu Tv .3 .

    f ((u)) = (v),

    Tv (x) = Tu(f (x)), f (x)j = xf (j), j [k].

    Tu(x) + Tv (y) = Tu(x + z), x , y U; z N(f (y))

    ( , , ).

    . ( ) 9: , CSclub, 2016 5 / 26

  • PCP A MAX-LCk(1, )

    1 (u, v) E (G ).2 Tu Tv .3 .

    f ((u)) = (v),

    Tv (x) = Tu(f (x)), f (x)j = xf (j), j [k].

    Tu(x) + Tv (y) = Tu(x + z), x , y U; z N(f (y))

    ( , , ).

    . ( ) 9: , CSclub, 2016 5 / 26

  • PCP A MAX-LCk(1, )

    1 (u, v) E (G ).2 Tu Tv .3 .

    f ((u)) = (v),

    Tv (x) = Tu(f (x)), f (x)j = xf (j), j [k].

    Tu(x) + Tv (y) = Tu(x + z), x , y U; z N(f (y))

    ( , , ).

    . ( ) 9: , CSclub, 2016 5 / 26

  • unsatG = 0, , A 1 .

    . : V , .

    Tu(x) = x(u), (v) = f ((u)) (uv) E .

    Tu(x) = x(u), Tv (y) = y(v), (v) = f ((u)).

    x(u) + yf ((u)) = x(u) + z(u) yf ((u)) = z(u),

    12(1 + ) = 1 .. ( ) 9: , CSclub, 2016 6 / 26

  • unsatG = 0, , A 1 .

    . : V , .

    Tu(x) = x(u), (v) = f ((u)) (uv) E .

    Tu(x) = x(u), Tv (y) = y(v), (v) = f ((u)).

    x(u) + yf ((u)) = x(u) + z(u) yf ((u)) = z(u),

    12(1 + ) = 1 .. ( ) 9: , CSclub, 2016 6 / 26

  • unsatG = 0, , A 1 .

    . : V , .

    Tu(x) = x(u), (v) = f ((u)) (uv) E .

    Tu(x) = x(u), Tv (y) = y(v), (v) = f ((u)).

    x(u) + yf ((u)) = x(u) + z(u) yf ((u)) = z(u),

    12(1 + ) = 1 .. ( ) 9: , CSclub, 2016 6 / 26

  • unsatG = 0, , A 1 .

    . : V , .

    Tu(x) = x(u), (v) = f ((u)) (uv) E .

    Tu(x) = x(u), Tv (y) = y(v), (v) = f ((u)).

    x(u) + yf ((u)) = x(u) + z(u) yf ((u)) = z(u),

    12(1 + ) = 1 .. ( ) 9: , CSclub, 2016 6 / 26

  • unsatG = 0, , A 1 .

    . : V , .

    Tu(x) = x(u), (v) = f ((u)) (uv) E .

    Tu(x) = x(u), Tv (y) = y(v), (v) = f ((u)).

    x(u) + yf ((u)) = x(u) + z(u) yf ((u)) = z(u),

    12(1 + ) = 1 .. ( ) 9: , CSclub, 2016 6 / 26

  • A < 1/2 1/2 + - . , 2 .

    1 S

    Tv (S)2

    ( ; ,Tv () = 0);

    2 j S ;3 (v) = j .

    . ( ) 9: , CSclub, 2016 7 / 26

  • A < 1/2 1/2 + - . , 2 .

    1 S

    Tv (S)2

    ( ; ,Tv () = 0);

    2 j S ;3 (v) = j .

    . ( ) 9: , CSclub, 2016 7 / 26

  • A < 1/2 1/2 + - . , 2 .

    1 S

    Tv (S)2

    ( ; ,Tv () = 0);

    2 j S ;3 (v) = j .

    . ( ) 9: , CSclub, 2016 7 / 26

  • A < 1/2 1/2 + - . , 2 .

    1 S

    Tv (S)2

    ( ; ,Tv () = 0);

    2 j S ;3 (v) = j .

    . ( ) 9: , CSclub, 2016 7 / 26

  • A < 1/2 1/2 + - . , 2 .

    1 S

    Tv (S)2

    ( ; ,Tv () = 0);

    2 j S ;3 (v) = j .

    . ( ) 9: , CSclub, 2016 7 / 26

  • A A = Tu, B = Tv 1/2 + uv . f 2uv .

    .

    ,

    E(uv)E

    [2uv ] >

    (E

    (uv)E[uv ]

    )2= 2.

    . ( ) 9: , CSclub, 2016 8 / 26

  • A A = Tu, B = Tv 1/2 + uv . f 2uv .

    .

    ,

    E(uv)E

    [2uv ] >

    (E

    (uv)E[uv ]

    )2= 2.

    . ( ) 9: , CSclub, 2016 8 / 26

  • S

    f

    i fodd(S)

    fodd(S)

    f odd(S) ={i :f 1(i) S },

    S .

    Pr[f ((u)) = (v)] >S 6=

    1|S |

    A(S)2B(f odd(S))2

    ( A = Tu, B = Tv .)

    . ( ) 9: , CSclub, 2016 9 / 26

  • S

    f

    i fodd(S)

    fodd(S)

    f odd(S) ={i :f 1(i) S },

    S .

    Pr[f ((u)) = (v)] >S 6=

    1|S |

    A(S)2B(f odd(S))2

    ( A = Tu, B = Tv .)

    . ( ) 9: , CSclub, 2016 9 / 26

  • Pr[f ((u)) = (v)] >S 6=

    1|S |

    A(S)2B(f odd(S))2

    1 S A(S)2.2 f odd(S) B

    B(f odd(S))2.3

    1/|S |. i f odd(S) j S , f (j) = i . , S A f odd(S) B

    jS

    1|S |

    if odd(S)

    1|f odd(S)|

    =1|S |

    .

    . ( ) 9: , CSclub, 2016 10 / 26

  • Pr[f ((u)) = (v)] >S 6=

    1|S |

    A(S)2B(f odd(S))2

    1 S A(S)2.2 f odd(S) B

    B(f odd(S))2.3

    1/|S |. i f odd(S) j S , f (j) = i . , S A f odd(S) B

    jS

    1|S |

    if odd(S)

    1|f odd(S)|

    =1|S |

    .

    . ( ) 9: , CSclub, 2016 10 / 26

  • Pr[f ((u)) = (v)] >S 6=

    1|S |

    A(S)2B(f odd(S))2

    1 S A(S)2.2 f odd(S) B

    B(f odd(S))2.3

    1/|S |. i f odd(S) j S , f (j) = i . , S A f odd(S) B

    jS

    1|S |

    if odd(S)

    1|f odd(S)|

    =1|S |

    .

    . ( ) 9: , CSclub, 2016 10 / 26

  • Pr[f ((u)) = (v)] >S 6=

    1|S |

    A(S)2B(f odd(S))2

    1 S A(S)2.2 f odd(S) B

    B(f odd(S))2.3

    1/|S |. i f odd(S) j S , f (j) = i . , S A f odd(S) B

    jS

    1|S |

    if odd(S)

    1|f odd(S)|

    =1|S |

    .

    . ( ) 9: , CSclub, 2016 10 / 26

  • S(f (x)) = f odd(S)(x) (: f (x)j = xf (j)).

    S(f (x)) =jS

    f (x)j =jS

    xf (j) =

    if odd(S)

    xi = f odd(S)(x),

    : x2 = 1 xi , |f 1(i) S | , 1.

    . ( ) 9: , CSclub, 2016 11 / 26

  • S(f (x)) = f odd(S)(x) (: f (x)j = xf (j)).

    S(f (x)) =jS

    f (x)j =jS

    xf (j) =

    if odd(S)

    xi = f odd(S)(x),

    : x2 = 1 xi , |f 1(i) S | , 1.

    . ( ) 9: , CSclub, 2016 11 / 26

  • S(f (x)) = f odd(S)(x) (: f (x)j = xf (j)).

    S(f (x)) =jS

    f (x)j =jS

    xf (j) =

    if odd(S)

    xi = f odd(S)(x),

    : x2 = 1 xi , |f 1(i) S | , 1.

    . ( ) 9: , CSclub, 2016 11 / 26

  • A, B > 1/2 + .

    S 6=A(S)2B(f odd(S))(1 2)|S | > 2.

    . .

    Pr[A(x)B(y) = A(x + z)] > 1/2 + ,

    E[A(x)B(y)A(x + z)] > 2.

    . ( ) 9: , CSclub, 2016 12 / 26

  • A, B > 1/2 + .

    S 6=A(S)2B(f odd(S))(1 2)|S | > 2.

    . .

    Pr[A(x)B(y) = A(x + z)] > 1/2 + ,

    E[A(x)B(y)A(x + z)] > 2.

    . ( ) 9: , CSclub, 2016 12 / 26

  • z = f (y) + w , w B ( ) :

    Ex ,y ,w

    [A(x)B(y)A(x + f (y) + w)] =

    = Ex ,y ,w

    [(S

    A(S)S(x))(

    U

    B(U)U(y))

    (

    T

    A(T )T (x + f (y) + w)))]

    =

    =S ,U,T

    A(S)B(U)A(T ) Ex

    [S(x)T (x)] Ey

    [U(y)T (f (y))] Ew

    [T (w)] =

    =S,U

    A(S)2B(U) Ey

    [U(y)S(f (y))] Ew

    [S(w)] > 2.

    . ( ) 9: , CSclub, 2016 13 / 26

  • z = f (y) + w , w B ( ) :

    Ex ,y ,w

    [A(x)B(y)A(x + f (y) + w)] =

    = Ex ,y ,w

    [(S

    A(S)S(x))(

    U

    B(U)U(y))

    (

    T

    A(T )T (x + f (y) + w)))]

    =

    =S,U,T

    A(S)B(U)A(T ) Ex

    [S(x)T (x)] Ey

    [U(y)T (f (y))] Ew

    [T (w)] =

    =S,U

    A(S)2B(U) Ey

    [U(y)S(f (y))] Ew

    [S(w)] > 2.

    . ( ) 9: , CSclub, 2016 13 / 26

    S(x + y) = S(x) S(y).

  • z = f (y) + w , w B ( ) :

    Ex ,y ,w

    [A(x)B(y)A(x + f (y) + w)] =

    = Ex ,y ,w

    [(S

    A(S)S(x))(

    U

    B(U)U(y))

    (

    T

    A(T )T (x + f (y) + w)))]

    =

    =S,U,T

    A(S)B(U)A(T ) Ex

    [S(x)T (x)] Ey

    [U(y)T (f (y))] Ew

    [T (w)] =

    =S,U

    A(S)2B(U) Ey

    [U(y)S(f (y))] Ew

    [S(w)] > 2.

    . ( )