multiway cut - rutgers university

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis

Modifying LPSolution

Multiway Cut

Sean Lowen

Franklin W. Olin College of Engineering

November 22, 2011

1 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Problem

Input

An undirected graph G = (V ,E ) with

◮ Non-negative weights we , ∀e ∈ E

◮ Set of terminals S = {s1, s2, · · · , sk} ⊆ V

Output

Subset of edges at min cost whose removal disconnects allterminals from each other

5

11

7

6

8

3

21

s2

s1

2 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Problem

Input




Output


5

11

7

6

8

3

21

s2

s1

5

11

7

6

8

3

21

s2

s1

Cost = 14 3 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Problem

Input




Output


5

11

7

6

8

3

21

s2

s1

5

11

7

6

8

3

21

s2

s1

Cost = 14

5

11

7

6

8

3

21

s2

s1

Cost = 6 4 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Outline

Simple 2-2/k Approximation

LP Relaxation

Understanding the LP Solution

Randomized Rounding Algorithm

Analysis

Modifying LP Solution

5 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Algorithm

Multiway Cut Algorithm

1. S = {s1, · · · , sk}

2. for i ← 1 to k do:ti ← contraction of S \ {si}Zi ← min si -ti cut

3. let w(Z1) ≤ w(Z2) ≤ · · · ≤ w(Zk)

4. return

k−1⋃

i=1Zi

6 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


2-2/k Approx Example

s2

s1

s3

1

5

2

3

4

2

5

2

6

41

7 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



s2

s1

s3

1

5

2

3

4

2

5

2

6

41

t1

Z1

1

5

2

3

4

25

2

64

s1

1

8 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



t1

Z1

1

5

2

3

4

25

2

64

s1

1

9 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



t1

Z1

1

5

2

3

4

25

2

64

s1

1

s2

t2

1

5

2

3

4

2

5

2

6

4

Z2

1

10 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



t1

Z1

1

5

2

3

4

25

2

64

s1

1

t3

s3

1

52

3

4

2

5

2

6

41

Z3

s2

t2

1

5

2

3

4

2

5

2

6

4

Z2

1

11 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



t1

Z1

1

5

2

3

4

25

2

64

s1

1

t3

s3

1

52

3

4

2

5

2

6

41

Z3

s2

t2

1

5

2

3

4

2

5

2

6

4

Z2

1

s2

s1

s3

Cost = 11

12 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Analysis

◮ A∗: optimal solution◮ A∗

i : edges in A∗ with exactly one endpoint in the partitioncontaining si

w(Z1) ≤ w(A∗

1)

w(Z2) ≤ w(A∗

2)

...

w(Zk) ≤ w(A∗

k)

A1

*

A3

*

A2

*

s1

s2

s3

w(⋃

i

Zi) ≤k∑

i=1w(A∗

i ) = 2 ·OPT

w(k−1⋃

i=1

Zi) ≤ (1− 1k)w(

⋃

i

Zi) ≤ 2(1− 1k) · OPT

13 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Analysis

◮ A∗: optimal solution◮ A∗

i : edges in A∗ with exactly one endpoint in the partitioncontaining si

w(Z1) ≤ w(A∗

1)

w(Z2) ≤ w(A∗

2)

...

w(Zk) ≤ w(A∗

k)

A1

*

A3

*

A2

*

s1

s2

s3

w(⋃

i

Zi) ≤k∑

i=1w(A∗

i ) = 2 ·OPT

w(k−1⋃

i=1

Zi) ≤ (1− 1k)w(

⋃

i

Zi) ≤ 2(1− 1k) · OPT

14 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Theorem

The above gives a (2− 2k)-approximation for multiway cut.

15 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Outline


LP Relaxation



Analysis


16 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation

◮ W.l.o.g., any feasible solution will partition the verticesinto C1,C2, · · · ,Ck

◮ si ∈ Ci

◮ δ(Ci ): set of edges leaving Ci

◮ vi =

{

1, if v ∈ Ci

0, otherwise

◮ z ie =

{

1, if e ∈ δ(Ci )

0, otherwise

◮ Obj: min⋃

i

w(δ(Ci ))

◮ In other words:

min 12

∑

e∈E

we

k∑

i=1z ie

Ci

Cj

u

v

si

sj

17 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation


◮ si ∈ Ci


◮ vi =

{

1, if v ∈ Ci

0, otherwise

◮ z ie =

{

1, if e ∈ δ(Ci )

0, otherwise

◮ Obj: min⋃

i

w(δ(Ci ))

◮ In other words:

min 12

∑

e∈E

we

k∑

i=1z ie

Ci

Cj

u

v

si

sj

18 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation


◮ si ∈ Ci


◮ vi =

{

1, if v ∈ Ci

0, otherwise

◮ z ie =

{

1, if e ∈ δ(Ci )

0, otherwise

◮ Obj: min⋃

i

w(δ(Ci ))

◮ In other words:

min 12

∑

e∈E

we

k∑

i=1z ie

Ci

Cj

u

si

sj

v

19 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation


◮ si ∈ Ci


◮ vi =

{

1, if v ∈ Ci

0, otherwise

◮ z ie =

{

1, if e ∈ δ(Ci )

0, otherwise

◮ Obj: min⋃

i

w(δ(Ci ))

◮ In other words:

min 12

∑

e∈E

we

k∑

i=1z ie

Ci

Cj

u

si

sj

v

uj= 0

ui = 1

vi = 0

vj = 1

20 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation


◮ si ∈ Ci


◮ vi =

{

1, if v ∈ Ci

0, otherwise

◮ z ie =

{

1, if e ∈ δ(Ci )

0, otherwise

◮ Obj: min⋃

i

w(δ(Ci ))

◮ In other words:

min 12

∑

e∈E

we

k∑

i=1z ie

Ci

Cj

u

e ze

i= |u - v|=1

si

sj

v

ui = 1

vi = 0

i i

21 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation


◮ si ∈ Ci


◮ vi =

{

1, if v ∈ Ci

0, otherwise

◮ z ie =

{

1, if e ∈ δ(Ci )

0, otherwise

◮ Obj: min⋃

i

w(δ(Ci ))

◮ In other words:

min 12

∑

e∈E

we

k∑

i=1z ie

Ci

Cj

u

e ze

i= |u - v|=1

si

sj

v

ui = 1

vi = 0

i i

22 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation


◮ si ∈ Ci


◮ vi =

{

1, if v ∈ Ci

0, otherwise

◮ z ie =

{

1, if e ∈ δ(Ci )

0, otherwise

◮ Obj: min⋃

i

w(δ(Ci ))

◮ In other words:

min 12

∑

e∈E

we

k∑

i=1z ie

Ci

Cj

u

e ze

i= |u - v|=1

si

sj

v

ui = 1

vi = 0

i i

23 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Constructing the LP

Integer Program

minimize 12

∑

e∈E

we

k∑

i=1z ie (1)

s.t.k∑

i=1

vi = 1, ∀v ∈ V (2)

z ie = |ui − vi |, ∀e = (u, v) ∈ E , 1 ≤ i ≤ k (3)sii = 1, ∀s ∈ S (4)vi ∈ {0, 1}, ∀v ∈ V , 1 ≤ i ≤ k (5)

24 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Constructing the LP

LP Relaxation

minimize 12

∑

e∈E

we

k∑

i=1

z ie (1)

s.t.k∑

i=1

vi = 1, ∀v ∈ V (2)

z ie = |ui − vi |, ∀e = (u, v) ∈ E , 1 ≤ i ≤ k (3)sii = 1, ∀s ∈ S (4)vi ≥ 0, ∀v ∈ V , 1 ≤ i ≤ k (5)

25 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Interpreting the LP

◮ xv = (x1v , x2v , ..., x

kv ), a point in R

k

◮ x iv ← vi

◮ all coordinates of xv sum to 1

◮ xv belongs to the (k − 1)-simplex

◮ xsi : unit vector with i th coordinate equal to 1

◮ Let de = 12

k∑

i=1z ie =

12

k∑

i=1|x iu − x iv |

26 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


∆3 Simplex

27 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Interpreting the LP

◮ xv = (x1v , x2v , ..., x

kv ), a point in R

k

◮ x iv ← vi

◮ all coordinates of xv sum to 1

◮ xv belongs to the (k − 1)-simplex

◮ xsi : unit vector with i th coordinate equal to 1

◮ Let de = 12

k∑

i=1z ie =

12

k∑

i=1|x iu − x iv |

28 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Constructing the LP

LP Relaxation

minimize 12

∑

e∈E

we

k∑

i=1

z ie

s.t.k∑

i=1vi = 1, ∀v ∈ V

z ie = |ui − vi |, ∀e = (u, v) ∈ E , 1 ≤ i ≤ k

sii = 1, ∀s ∈ S

vi ≥ 0, ∀v ∈ V , 1 ≤ i ≤ k

29 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation

LP

minimize∑

e∈E

wede

s.t.k∑

i=1x iv = 1, ∀v ∈ V

de =12

k∑

i=1|x iu − x iv | ∀e = (u, v) ∈ E

x isi = 1, ∀s ∈ S

x iv ≥ 0, ∀v ∈ V , 1 ≤ i ≤ k

30 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation

Is this a valid LP relaxation?

ws2

s1

vu

s3

2 2

1

1 12

22

2

31 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation


ws2

s1

vu

s3

2 2

1

1 12

22

2

ws2

s1

vu

s3

2 2

1

1 12

22

2

32 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation


ws2

s1

vu

s3

2 2

1

1 12

22

2

ws2

s1

vu

s3

ws2

s1

vu

s3

2 2

1

1 12

22

2

33 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation


ws2

s1

vu

s3

2 2

1

1 12

22

2

ws2

s1

vu

s3

ws2

s1

vu

s3

2 2

1

1 12

22

2

xw

s2

s1

xv

xu s

3

(0,0,1)(0,1,0)

(1,0,0)

2

2

2

1

1

34 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Relaxation

LP

minimize∑

e∈E

wede

s.t.k∑

i=1x iv = 1, ∀v ∈ V

de =12

k∑

i=1|x iu − x iv | ∀e = (u, v) ∈ E

x isi = 1, ∀s ∈ S

x iv ≥ 0, ∀v ∈ V , 1 ≤ i ≤ k

35 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Outline


LP Relaxation



Analysis


36 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


LP Solution: Example

ws2

s1

vu

s3

2 2

1

1 12

22

2

OPT = 8

37 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



ws2

s1

vu

s3

2 2

1

1 12

22

2

OPT = 8

xw

s2

s1

xv

xu

s3

(1,0,0)

(.5,0,.5)

(0,0,1)(0,.5,.5)(0,1,0)

(.5,.5,0)

LP-OPT = 7.5

38 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



ws2

s1

vu

s3

2 2

1

1 12

22

2

OPT = 8

xw

s2

s1

xv

xu

s3

(1,0,0)

(.5,0,.5)

(0,0,1)(0,.5,.5)(0,1,0)

(.5,.5,0)

LP-OPT = 7.5

39 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



ws2

s1

vu

s3

2 2

1

1 12

22

2

OPT = 8

xw

s2

s1

xv

xu

s3

(1,0,0)

(.5,0,.5)

(0,0,1)(0,.5,.5)(0,1,0)

(.5,.5,0)

LP-OPT = 7.5

40 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Outline


LP Relaxation



Analysis


41 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Notation

◮ Assume that the LP gives a solution in which, for anyedge (u,v), its endpoints differ in at most two coordinates.

◮ Ei : edges whose endpoints differ in coordinate i .

◮ Wi :∑

e∈Ei

wede

◮ B(si , ρ): {v ∈ V |x iv ≥ ρ}.

◮ Ci : partition containing si

42 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Algorithm

Algorithm

1. Compute optimum LP solution.

2. Renumber the terminals so that W1 ≤ · · · ≤Wk

3. Pick uniformly at random ρ ∈ (0, 1) andσ ∈ {(1, 2, . . . , k − 1, k), (k − 1, k − 2, . . . , 1, k)}.

4. for i = 1 to k − 1:Cσ(i)

← B(si , ρ)−⋃

j<i

Cσ(j)

5. Ck ← all remaining vertices

6. C : set of edges that run between sets C1, . . . ,Ck

7. return C

43 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Example of Algorithm Partitions

ρ < 12 , σ = (1, 2, 3)

44 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



ρ < 12 , σ = (1, 2, 3)

45 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



ρ < 12 , σ = (1, 2, 3)

46 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



ρ < 12 , σ = (1, 2, 3)

47 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



ρ < 12 , σ = (1, 2, 3)

48 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Outline


LP Relaxation



Analysis


49 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Analysis

Lemma

If e ∈ E − Ek ,Pr[e ∈ C ] ≤ 1.5deIf e ∈ Ek ,Pr[e ∈ C ] ≤ de

◮ Consider edge e = (u, v), where u and v differ incoordinates i and j

◮ By definition, x iu − x iv = xjv − x

ju = de

◮ Edge e will only be cut if xu and xv end up in differentpartitions, i.e., ρ falls in either the (x iu , x

iv ) or (x

jv , x

ju)

intervals.

◮ Let’s call those intervals α and β, respectively.

50 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Analysis

Lemma




ju = de


iv ) or (x

jv , x

ju)

intervals.


51 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Analysis

Lemma




ju = de


iv ) or (x

jv , x

ju)

intervals.


52 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Analysis

Lemma




ju = de


iv ) or (x

jv , x

ju)

intervals.


53 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Analysis: e ∈ E \ Ek

Edge e = (u, v) differs in coordinates i & j

Pr[e ∈ C ] = Pr[ρ ∈ (α ∪ β)] = |α|+ |β| ≤ 2de

54 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case I: ρ ∈ β and σ = (. . . , j , . . . , i , . . .)

55 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case I: ρ ∈ β and σ = (. . . , j , . . . , i , . . .)

56 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case II: ρ ∈ β and σ = (. . . , i , . . . , j , . . .)

57 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case II: ρ ∈ β and σ = (. . . , i , . . . , j , . . .)

58 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case II: ρ ∈ β and σ = (. . . , i , . . . , j , . . .)

59 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case II: ρ ∈ β and σ = (. . . , i , . . . , j , . . .)

60 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case III: ρ ∈ α and σ = (. . . , i , . . . , j , . . .)

61 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case III: ρ ∈ α and σ = (. . . , i , . . . , j , . . .)

62 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case IV: ρ ∈ α and σ = (. . . , j , . . . , i , . . .)

63 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Case IV: ρ ∈ α and σ = (. . . , j , . . . , i , . . .)

64 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis



Pr[e ∈ C ] = Pr[ρ ∈ β] + Pr[(ρ ∈ α) ∧ (σ = (. . . , i , . . . , j , . . .))]

= |β|+|α|

2≤ 1.5de

Lemma

If e ∈ E − Ek ,Pr[e ∈ C ] ≤ 1.5de

65 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Analysis: e ∈ Ek

An edge in Ek can only be cut if ρ ∈ β.

◮ σ always = (. . . , i , . . . , k)

◮ k gets all remaining vertices

◮ Pr[e ∈ C ] = |β| = de

66 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Analysis: e ∈ Ek

An edge in Ek can only be cut if ρ ∈ β.

◮ σ always = (. . . , i , . . . , k)

◮ k gets all remaining vertices

◮ Pr[e ∈ C ] = |β| = de

Lemma

If e ∈ Ek ,Pr[e ∈ C ] ≤ de

67 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Approximation

OPT f =∑

e∈E

wede

Note that every edge belongs to exactly 2 of the Ei sets.

k∑

i=1

Wi = 2 ·OPT f

Since Wk is the largest of the sets:

Wk =∑

e∈Ek

wede ≥2k· OPTf

68 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Approximation

OPT f =∑

e∈E

wede


k∑

i=1

Wi = 2 ·OPT f


Wk =∑

e∈Ek

wede ≥2k· OPTf

69 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Approximation

OPT f =∑

e∈E

wede


k∑

i=1

Wi = 2 ·OPT f


Wk =∑

e∈Ek

wede ≥2k· OPTf

70 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Approximation

E[w(C )] =∑

e∈E

wePr[e ∈ C ]

=∑

e∈E−Ek

wePr[e ∈ C ] +∑

e∈Ek

wePr[e ∈ Ek ]

≤ 1.5∑

e∈E−Ek

wede +∑

e∈Ek

wede

= 1.5∑

e∈E

wede − 0.5∑

e∈Ek

wede

= 1.5∑

e∈E

wede − 0.5Wk

≤ (1.5 − 1/k) ·OPT f

71 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Conclusion

Theorem

There is a (1.5− 1k)-approximation for multiway cut.

72 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Outline


LP Relaxation



Analysis


73 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Modifying the LP

Convert the LP solution to another LP solution of same cost inwhich each edge differs in at most two coordinates.

◮ xu & xv differ in morethan two coordinates.

◮ i : coordinate in which xuand xv differ the least.Let x iv > x iu.

◮ Let α = x iv − x iu

◮ There is a coordinate j

s.t. x ju − xjv ≥ α

◮ ∀k 6= i , j , xkw ← xkv

◮ wuw ,wwv ← wuv

xu

xu( ),... , ......

xu

xv

wuv

, ,i j

xv( ),... , ...... xv

i j

, ,

74 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Modifying the LP







◮ ∀k 6= i , j , xkw ← xkv


xu

xu( ),... , ......

xu

xv

wuv

, ,i j

xv( ),... , ...... xv

i j

, ,

75 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Modifying the LP







◮ ∀k 6= i , j , xkw ← xkv


xu

xu( ),... , ......

xu

xv

wuv

, ,i j

xv( ),... , ...... xv

i j

, ,

76 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Modifying the LP







◮ ∀k 6= i , j , xkw ← xkv


xu

xu( ),... , ......

xu

xv

wuv

, ,i j

xv( ),... , ...... xv

i j

, ,

77 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Modifying the LP







◮ ∀k 6= i , j , xkw ← xkv


wuv

xu

xu( ),... , ......

xu

xw

xv

,

xv( ),... , ...... xv

i j

, ,

,i j

78 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Modifying the LP







◮ ∀k 6= i , j , xkw ← xkv


xv

xu( ),... , ...... + α, ,

wuv

wuv

xu

xu( ),... , ......

xu

xw

xv

, ,i j i j

xv( ),... , ...... xv

i j

, ,

79 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Modifying the LP







◮ ∀k 6= i , j , xkw ← xkv


xv

xu( ),... , ...... + α, ,

wuv

wuv

xu

xu( ),... , ......

xu

xw

xv

, ,i j i j

xv( ),... , ...... xv

i j

, ,

80 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


Does wuvduv = wuwduw + wwvdwv?

wuvduv?= wuvduw + wuvdwv

duv?= duw + dwv

Let duv = σ

duw = σ − 2α

dwv = 2α

duw + dwv = 2α+ σ − 2α

duw + dwv = σ = duv

xv

xu( ),... , ...... + α, ,

wuv

wuv

xu

xu( ),... , ......

xu

xw

xv

, ,i j i j

xv( ),... , ...... xv

i j

, ,

81 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis




duv?= duw + dwv

Let duv = σ

duw = σ − 2α

dwv = 2α



xv

xu( ),... , ...... + α, ,

wuv

wuv

xu

xu( ),... , ......

xu

xw

xv

, ,i j i j

xv( ),... , ...... xv

i j

, ,

82 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis




duv?= duw + dwv

Let duv = σ

duw = σ − 2α

dwv = 2α



xv

xu( ),... , ...... + α, ,

wuv

wuv

xu

xu( ),... , ......

xu

xw

xv

, ,i j i j

xv( ),... , ...... xv

i j

, ,

83 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis




duv?= duw + dwv

Let duv = σ

duw = σ − 2α

dwv = 2α



xv

xu( ),... , ...... + α, ,

wuv

wuv

xu

xu( ),... , ......

xu

xw

xv

, ,i j i j

xv( ),... , ...... xv

i j

, ,

84 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis




duv?= duw + dwv

Let duv = σ

duw = σ − 2α

dwv = 2α



xv

xu( ),... , ...... + α, ,

wuv

wuv

xu

xu( ),... , ......

xu

xw

xv

, ,i j i j

xv( ),... , ...... xv

i j

, ,

85 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis




duv?= duw + dwv

Let duv = σ

duw = σ − 2α

dwv = 2α



xv

xu( ),... , ...... + α, ,

wuv

wuv

xu

xu( ),... , ......

xu

xw

xv

, ,i j i j

xv( ),... , ...... xv

i j

, ,

86 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis




duv?= duw + dwv

Let duv = σ

duw = σ − 2α

dwv = 2α



xv

xu( ),... , ...... + α, ,

wuv

wuv

xu

xu( ),... , ......

xu

xw

xv

, ,i j i j

xv( ),... , ...... xv

i j

, ,

Lemma

There is a way to modify the LP s.t. for any edge e = (u, v),xu and xv differ in 0 or 2 coordinates, at no additional cost.

87 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


References

G. Calinescu, H. Karloff, and Y. Rabani. An Improved

Approximation Algorithm for Multiway Cut. Proc. of 30thACM Symposium on the Theory of Computing (STOC’98),1998.

V. Vazirani. Approximation Algorithms. Berlin: Springer, 2003.

D. Williamson, D. Shmoys. The Design of Approximation

Algorithms. New York: Cambridge University Press, 2011.

88 / 89

Multiway Cut

Sean Lowen

2-2/k Approx

Simplex

UnderstandingLP

Randomization

Analysis


End

Thank you!

89 / 89

multiway cut - rutgers university

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