# 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.

. ( )