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S:e∈δ(S) y _S ≤ c e

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• (c _uv − ^P S:uv∈δ(S) y _S )/2

^F4

u

v

b4b]b{8]wxowxb_4]P{ZL{x

• c uv − ^P S:uv∈δ(S) y _S

⁴

u

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v

a{xFvxwz{Z%wL b4b]%{8|{8]wxowxb_4{ZL{

• ∞

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u

v

b4b]b{8]wxowxb_4]F{ZL{x

!

{Z4]{ f4bF{]4]wx}~{&b}~w{Z}#+1wxzw F{Z{4bb}|!~{ b

F

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b |{ ]wxo&wxb4:{Zx{: 4b >wx}#+{q !{ b4 4wf{xz{xz{Z4]{xz a4bF{x

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=b}~w84]wx4b{&>k{x]wz{8a|F{Z}~{xLF]4x{Z}~wx

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!

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R

⁺

1 F ← (V G

∅ ) 2 E ext ← E G

3 y _S ← 0

F{Z{84wfwo]wx4C/|4w

S

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4

bK*|F{Z_4w

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8 y _S ← y _S +

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F 9 E F ← E F ∪ { e }

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v

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F } 11

⁴ ^C4{

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R

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12

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2

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* z]wxzw

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5 wx|

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L {ZzwLzwL{Z{C*_|@*_|)w{Z}#+1wxzw-,wfb{ZP(. }}{Zz4wx *_| {x {ZF{ZzwL)1b<| {

b1wx}1q|{

R

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c(F ⁰ ) ≤

2 − 2 t

;)

(G, c, R ) ,

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wx

F ⁰

.{NDFwx4]{b1wx}f~{&b}~w {Z}#+1wxzw

;)

(G, c, R )

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t = ^P _T _∈R | T |

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uv

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p

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S:uv∈δ(S)

y _S = ^X

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y _S ⁰ +

(uv) +

(uv)

= ^X

S:uv∈δ(S)

y _S ⁰ + 2 (c uv − ^P S:uv∈δ(S) y ⁰ _S )

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(uv)

= ^X

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y ⁰ _S +



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

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c(F ⁰ ) = ^X

e∈E _F 0

c e = ^X

e∈E _F 0

X

S:e∈δ(S)

y _S = ^X

S

| δ F ⁰ (S) | y _S ,

⁸ ^"!^7;

wx

δ F ⁰ (S)

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2 − 2

t

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y _S ≤

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| δ _F 0 (S) | y _S ≤

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X

S

y _S .

⁸ ^"!^* ^;

(15)

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θ

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θ ^P _S∈S _F | δ F ⁰ (S) |

1bK*|F{Z_4w *_|Rw}{xwa4b4w4b J{Z|zb_{xw

2 − ² _t |S F | θ

^!

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X

S∈S F

| δ F ⁰ (S) | ≤

2 − 2 t

|S ^F |

. {Z{Zzb_4)~44w *_|4bzwL74w x{Z<w

{zUWQSX d

%'

+ % + %$ %882

X

S∈S F

| δ _F 0 (S) | ≤

2 − 2 t

|S ^F | .

5

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S

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n|{ZDFwx4]4{ |

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| δ _F 0 (S) | ≥ 1

⁴ ^C{

F ₀

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! +1wx{a ]wx~b4qw@+L{t>w

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S ⁰

vxw{xC4{x]b4]b

F ₀ }

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ta{Z&bzwL *_| Fvxw a~4bb4b|4wLb

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!

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| E H | ≤ | V H | − 1 = |S F | + |Z F | − 1.

ywxzwx|4w:}{xw {Z&bzwL *|

P

S∈V H | δ H (S) | = 2 | E H |

^L {x4n]wxzw

| δ F ⁰ (S) | ≤ | δ H (S) |

^F{Z{

4wfw

S

^b

V H

4bzwL *|

X

S∈V H

| δ F ⁰ (S) | ≤ 2 | E H | ≤ 2( |S F | + |Z F | − 1)

⇒ ^X

S∈S F ∪Z F

| δ F ⁰ (S) | ≤ 2( |S F | + |Z F | − 1)

⇒ ^X

S∈S F

| δ F ⁰ (S) | ≤ 2( |S ^F | + |Z ^F | − 1) − ^X

S∈Z F

| δ F ⁰ (S) | .

(16)

ywxnFwx4:*_|L]wxzw

F ⁰

^. ^|{

R

^O DFwx4]{zo{Z}FFvxw f~44

S

^b

Z ^F

^{Z} ^*_|

| δ F ⁰ (S) | = 1

L]44zw_w

| δ _F 0 (S) | ≥ 2

F{Z{84wfw

S

^b

Z F

!

MDw4+1w

2( |S F | + |Z F | − 1) − ^X

S∈Z F

| δ F ⁰ (S) | ≤ 2( |S F | + |Z F | − 1) − 2 |Z F | = 2 |S F | − 2

La&wx{Z4w

X

S∈S F

| δ _F 0 (S) | ≤ 2 |S F | − 2 =

2 − 2

|S F |

|S F | ≤

2 − 2 t

|S F | ,

]wxzw9*|b !{ZzwL4w L{Z

!

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2 − ² t

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5

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