Приближенное решение задач комбинаторной оптимизации: алгоритмы и трудность, осень 2016: Задача об относительном разрезе. Теорема Бургейна

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• :

3: .

.

. ..

-, Computer Science Club, 2016

. ( ) 3: Sparsest cut , CSclub, 2016 1 / 34

• (S ,V \ S), S V , G (V ,E )

rG (S) =E (S ,V \ S)|S | |V \ S |

,

( ( S , S)).

(the sparsest cut problem)

: G (V ,E )

:

rG = minSV

rG (S).

. ( ) 3: Sparsest cut , CSclub, 2016 2 / 34

• (S ,V \ S), S V , G (V ,E )

rG (S) =E (S ,V \ S)|S | |V \ S |

,

( ( S , S)).

(the sparsest cut problem)

: G (V ,E )

:

rG = minSV

rG (S).

. ( ) 3: Sparsest cut , CSclub, 2016 2 / 34

• :

, .

hE (G ) = minSV

hE (S ,V \ S);

hE (S ,V \ S) =E (S ,V \ S)

d min(|S |, |V \ S |).

d- n

d

n hE (G ) 6 rG 6

2d

n hE (G ).

. ( ) 3: Sparsest cut , CSclub, 2016 3 / 34

• :

, .

hE (G ) = minSV

hE (S ,V \ S);

hE (S ,V \ S) =E (S ,V \ S)

d min(|S |, |V \ S |).

d- n

d

n hE (G ) 6 rG 6

2d

n hE (G ).

. ( ) 3: Sparsest cut , CSclub, 2016 3 / 34

• :

:

, 1/d ( , ).

G 1 2 = 1 2 .

d- n

d

n G 6 rG 6

22 dn

G .

22/G ( n).

. ( ) 3: Sparsest cut , CSclub, 2016 4 / 34

• :

:

, 1/d ( , ).

G 1 2 = 1 2 .

d- n

d

n G 6 rG 6

22 dn

G .

22/G ( n).

. ( ) 3: Sparsest cut , CSclub, 2016 4 / 34

• :

:

, 1/d ( , ).

G 1 2 = 1 2 .

d- n

d

n G 6 rG 6

22 dn

G .

22/G ( n).

. ( ) 3: Sparsest cut , CSclub, 2016 4 / 34

• : .

: ,

, .

(non-uniform sparsest cut)

.

1, 0; 1.

. ( ) 3: Sparsest cut , CSclub, 2016 5 / 34

• G (V ,EG ), H(V ,EH) (S ,V \ S)

rG ,H(S) =EG (S ,V \ S)EH(S ,V \ S)

.

,

, ,

{0, 1}.

: G (V ,EG ), H(V ,EH) .

:

rG ,H = minSV

rG ,H(S).

. ( ) 3: Sparsest cut , CSclub, 2016 6 / 34

• G (V ,EG ), H(V ,EH) (S ,V \ S)

rG ,H(S) =EG (S ,V \ S)EH(S ,V \ S)

.

,

, ,

{0, 1}.

: G (V ,EG ), H(V ,EH) .

:

rG ,H = minSV

rG ,H(S).

. ( ) 3: Sparsest cut , CSclub, 2016 6 / 34

• : (0, 1)-

(S ,V \ S):

dS(u, v) = |S(u) S(v)|, S(v) =

{1, v S ,0, .

(dS(u, v))u,vV (0, 1)- n2- , n = |V |.

n- .

EG (S ,V \ S) =

(u,v)E(G)

dS(u, v).

. ( ) 3: Sparsest cut , CSclub, 2016 7 / 34

• : (0, 1)-

(S ,V \ S):

dS(u, v) = |S(u) S(v)|, S(v) =

{1, v S ,0, .

(dS(u, v))u,vV (0, 1)- n2- , n = |V |.

n- .

EG (S ,V \ S) =

(u,v)E(G)

dS(u, v).

. ( ) 3: Sparsest cut , CSclub, 2016 7 / 34

• : (0, 1)-

(S ,V \ S):

dS(u, v) = |S(u) S(v)|, S(v) =

{1, v S ,0, .

(dS(u, v))u,vV (0, 1)- n2- , n = |V |.

n- .

EG (S ,V \ S) =

(u,v)E(G)

dS(u, v).

. ( ) 3: Sparsest cut , CSclub, 2016 7 / 34

• : 0 6 dS(u, v) 6 1, d(u, v) = d(v , u).

?

: ,

. k = 2, 3, . . . , .

k = 2

. , dS(u, u) = 0. dS(u, v), u 6= v . (0, 1)-.

. ( ) 3: Sparsest cut , CSclub, 2016 8 / 34

• : 0 6 dS(u, v) 6 1, d(u, v) = d(v , u).

?

: ,

. k = 2, 3, . . . , .

k = 2

. , dS(u, u) = 0. dS(u, v), u 6= v . (0, 1)-.

. ( ) 3: Sparsest cut , CSclub, 2016 8 / 34

• : 0 6 dS(u, v) 6 1, d(u, v) = d(v , u).

?

: ,

. k = 2, 3, . . . , .

k = 2

. , dS(u, u) = 0. dS(u, v), u 6= v . (0, 1)-.

. ( ) 3: Sparsest cut , CSclub, 2016 8 / 34

• :

dS(u, v) + dS(v ,w) > dS(u,w).

.

d : V V R , u, v ,w V

d(u, v) > 0,

d(u, u) = 0,

d(u, v) = d(v , u),

d(u, v) + d(v ,w) > d(u,w).

. ( ) 3: Sparsest cut , CSclub, 2016 9 / 34

• :

dS(u, v) + dS(v ,w) > dS(u,w).

.

d : V V R , u, v ,w V

d(u, v) > 0,

d(u, u) = 0,

d(u, v) = d(v , u),

d(u, v) + d(v ,w) > d(u,w).

. ( ) 3: Sparsest cut , CSclub, 2016 9 / 34

• :

dS(u, v) + dS(v ,w) > dS(u,w).

.

d : V V R , u, v ,w V

d(u, v) > 0,

d(u, u) = 0,

d(u, v) = d(v , u),

d(u, v) + d(v ,w) > d(u,w).

. ( ) 3: Sparsest cut , CSclub, 2016 9 / 34

• :

rG ,H = min6=SV

(u,v)EG dS(u, v)(u,v)EH dS(u, v)

.

:

LRG ,H = mind(,)M

(u,v)EG d(u, v)(u,v)EH d(u, v)

.

M .

?

. ( ) 3: Sparsest cut , CSclub, 2016 10 / 34

• :

rG ,H = min6=SV

(u,v)EG dS(u, v)(u,v)EH dS(u, v)

.

:

LRG ,H = mind(,)M

(u,v)EG d(u, v)(u,v)EH d(u, v)

.

M .

?

. ( ) 3: Sparsest cut , CSclub, 2016 10 / 34

• :

rG ,H = min6=SV

(u,v)EG dS(u, v)(u,v)EH dS(u, v)

.

:

LRG ,H = mind(,)M

(u,v)EG d(u, v)(u,v)EH d(u, v)

.

M .

?

. ( ) 3: Sparsest cut , CSclub, 2016 10 / 34

• :

d(, ) .

LRG ,H (u,v)EG

du,v min(u,v)EH

du,v = 1,

du,v 6 du,w + dw ,v , u, v ,w V ,du,v > 0, u, v V .

du,v .

. ( ) 3: Sparsest cut , CSclub, 2016 11 / 34

• C G , H

LRG ,H 6 rG ,H 6 C log |V | LRG ,H .

:

.

:

1 ,

`1-.

2 V `1-, , O(log |V |) .( .)

. ( ) 3: Sparsest cut , CSclub, 2016 12 / 34

• C G , H

LRG ,H 6 rG ,H 6 C log |V | LRG ,H .

:

.

:

1 ,

`1-.

2 V `1-, , O(log |V |) .( .)

. ( ) 3: Sparsest cut , CSclub, 2016 12 / 34

• C G , H

LRG ,H 6 rG ,H 6 C log |V | LRG ,H .

:

.

:

1 ,

`1-.

2 V `1-, , O(log |V |) .( .)

. ( ) 3: Sparsest cut , CSclub, 2016 12 / 34

• C G , H

LRG ,H 6 rG ,H 6 C log |V | LRG ,H .

:

.

:

1 ,

`1-.

2 V `1-, , O(log |V |) .( .)

. ( ) 3: Sparsest cut , CSclub, 2016 12 / 34

• `1-

`1- Rm

(x1, . . . , xm) (y1, . . . , ym)1 =mi=1

|xi yi |.

`1-:

d : V V R, f : V Rm,

d(u, v) = f (u) f (v)1 u, v V .

. ( ) 3: Sparsest cut , CSclub, 2016 13 / 34

• `1-

`1- Rm

(x1, . . . , xm) (y1, . . . , ym)1 =mi=1

|xi yi |.

`1-:

d : V V R, f : V Rm,

d(u, v) = f (u) f (v)1 u, v V .

. ( ) 3: Sparsest cut , CSclub, 2016 13 / 34

• `1-

`1- Rm

(x1, . . . , xm) (y1, . . . , ym)1 =mi=1

|xi yi |.

`1-:

d : V V R, f : V Rm,

d(u, v) = f (u) f (v)1 u, v V .

. ( ) 3: Sparsest cut , CSclub, 2016 13 / 34

`1-, R1

dS(u, v) = |S(u) S(v)|.

• `1-

r1G ,H `1-.

`1-

G (V ,EG ), H(V ,EH) r1G ,H = rG ,H .

, ,

f : V Rd S , S `1-, f .

. ( ) 3: Sparsest cut , CSclub, 2016 14 / 34

• , x1, . . . xn y1, . . . , yn

x1 + + xny1 + + yn

> mini

xiyi.

,

.

. ( ) 3: Sparsest cut , CSclub, 2016 15 / 34

• , x1, . . . xn y1, . . . , yn

x1 + + xny1 + + yn

> mini

xiyi.

,

.

. ( ) 3: Sparsest cut , CSclub, 2016 15 / 34

• `1-

r1G ,H 6 rG ,H ,

`1-.

r1G ,H R1.

f : V Rm fi : v 7 f (v)i :(u,v)EG f (u) f (v)1(u,v)EH f (u) f (v)1

=

i

(u,v)EG |fi (u) fi (v)|

i

(u,v)EH |fi (u) fi (v)|

>

> mini

(u,v)EG |fi (u) fi (v)|(u,v)EH |fi (u) fi (v)|

.

. ( ) 3: Sparsest cut , CSclub, 2016 16 / 34

• `1-

r1G ,H 6 rG ,H ,

`1-.

r1G ,H R1.

f : V Rm fi : v 7 f (v)i :(u,v)EG f (u) f (v)1(u,v)EH f (u) f (v)1

=

i

(u,v)EG |fi (u) fi (v)|

i

(u,v)EH |fi (u) fi (v)|

>

> mini

(u,v)EG |fi (u) fi (v)|(u,v)EH |fi (u) fi (v)|

.

. ( ) 3: Sparsest cut , CSclub, 2016 16 / 34

• `1-

r1G ,H 6 rG ,H ,

`1-.

r1G ,H R1.

f : V Rm fi : v 7 f (v)i :(u,v)EG f (u) f (v)1(u,v)EH f (u) f (v)1

=

i

(u,v)EG |fi (u) fi (v)|

i

(u,v)EH |fi (u) fi (v)|

>

> mini

(u,v)EG |fi (u) fi (v)|(u,v)EH |fi (u) fi (v)|

.

. ( ) 3: Sparsest cut , CSclub, 2016 16 / 34

• 1- `1-

1- g : V R S , ,

g .

t [min g(v);max g(v)].

. ( ) 3: Sparsest cut , CSclub, 2016 17 / 34

• 1- `1-

1- g : V R S , ,

g .

St = {v V : g(v) 6 t}

min g(v) max g(v)t

St

t [min g(v);max g(v)].

. ( ) 3: Sparsest