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