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On MultiCuts and Related Problems Michael Langberg California Institute of Technology Joint work with Adi Avidor. This talk. Part I: Generalization of both Min. MultiCut and Min. Multiway Cut problems. Part II: Minimum Uncut problem. Part I: Minimum MultiCut. Input: G=(V,E) .

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### On MultiCuts andRelated ProblemsMichael LangbergCalifornia Institute of TechnologyJoint work with Adi Avidor

This talk
• Part I:

Generalization of both Min.MultiCut and Min.Multiway Cut problems.

• Part II:

Minimum Uncut problem.

Part I: Minimum MultiCut
• Input:
• G=(V,E).
• : E  R+.
• {(si,ti)}i=1..k.
• Objective:
• E’  E that disconnect

sifrom ti for all i=1..k.

• Measure:E’ of minimum weight.

s1

t3

s3

MultiCut

t2

t1

s2

G=(V,E)

Minimum Multiway Cut
• Input:
• G=(V,E).
• : E  R+.
• {s1,s2,…,sk}.
• Objective:
• E’  E that disconnect

sifrom sj.

• Measure:E’ of minimum weight.

s1

s4

s3

Multiway Cut

s5

s6

s2

G=(V,E)

Multicut vs. Multiway cut.
• Multicut: disconnect pairs {si,ti}i=1 .. k.
• MultiwayCut: disconnect {s1,s2,…,sk}.
• NP-hard, extensively studied in the past.
• Will present known results shortly.
• Roughly:
• Multiway Cut < Multicut.
• Mutiway Cut: constant app.
• Multicut: only logarithmic app. is known.
Our generalization: Minimum Multi-Multiway Cut
• Input:
• G=(V,E).
• : E  R+.
• {S1,S2,…,Sk}: Si V.
• Objective:
• E’  E that disconnect

all vertices in Si for i=1..k.

• Measure:E’ of minimum weight.

S1

s13

s11

S3

s12

s21

s21

S2

s24

s21

s23

s22

G=(V,E)

Why generalization?
• Input:
• G=(V,E).
• : E  R+.
• {S1,S2,…,Sk}: Si V.
• Multicut ({(si,ti)}i=1..k)
• Each setSi={si,ti}.
• Multiway Cut:({s1,s2,…,sk})
• Singe set S1 of size k.

s13

s11

s12

s21

s21

s24

s21

s23

s22

G=(V,E)

Previous results

+ Our results

“Light inst.”

log(Opt)loglog(Opt)

[Seymore,Even et al.]

Multicut

APX-Hard

[Dahlhaus et al.]

O(log(k))

[Garg et al.]

{(si,ti)}i=1..k

1.34 - k

[Cainescu et al.

Karger et al,

CunninghamTang]

Multiway

Cut

APX-Hard

[Dahlhaus et al.]

---

{s1,s2,…,sk}

Multi-

Multiway

Cut

“Light inst.”

O(log(Opt))

APX-Hard

[Dahlhaus et al]

{S1,S2,…,Sk}

O(log(k))

Results and proof techniques
• Multi-Multiway Cut results:
• 4ln(k+1) approximation.
• 4ln(2OPT) app. (edge weights  1).
• Proof:
• Natural LP relaxation.
• Rounding: variation of region growing tech. [LeightonRao, Klein et al., Garg et al.]
LP

Multi-Multiway Cut

LP:

Min:e(e)x(e)

st:

For every path P we

want to disconnect

• ePx(e)1

x(e)0

• Correctness: x(e){0,1}

s13

s11

P

s12

s21

s21

s24

P

s21

s23

s22

G=(V,E)

Rounding – region growing

Multi-Multiway Cut

• From LP: obtained fractional edge values.
• Implies a semi-metric on G.
• Simultaneously grow balls around vertices of connected sets until certain criteria.
• Each ball containes vertices close to center.
• Remove all edges cut by balls.

P1

s13

s11

P2

s12

s21

s21

s24

s21

s23

s22

G=(V,E)

Central: define the stopping criteria

Stopping criteria + analysis
• Based on that introduced by [GargVaziraniYannakakis].
• Consider both volume and cut value of union of balls.
• Main differences:
• Simultaneously grow balls.
• log(Opt):
• Change volume definition.
• Grow large balls only.

Multi-Multiway Cut

P1

s13

s11

P2

s12

s21

s21

s24

s21

s23

s22

G=(V,E)

Part II: Minimum Uncut
• Input:
• G=(V,E); : E  R+.
• Objective:
• Cut
• Measure:
• Minimum weight of uncut edges (dual to Min. Cut).
• Find subset E’ of E of minimum weight s.t. G=(V,E-E’) bipartite.

Cut

G=(V,E)

Min. Uncut: previous results
• Min-Uncut < Min. MultiCut

[KleinRaoAgrawalRavi].

• App. ratio of O(log(|V|)).
• Remainder of this talk: observations on attempt to improve app. ratio.

G=(V,E)

Observations
• Our results imply:

O(log(Opt))approximation:

If an undirected graph G can be made bipartite by the deletion of W edges, then a set of O(W log W) edges whose deletion makes the graph bipartite can be efficiently found.

• Min-Uncut < Min. MultiCut
Observations: LP
• Recall: Min. uncut has ratio O(log(n)).
• Can show:
• Natural LP has IG (log(n)).
• LP enhanced with “triangle” constraints: IG (log(n)).
• LP enhanced with “odd cycle” con.: IG (log(n)).
• LP combined with both:IGnot resolved.

x=1 x=0 x=1

1-x = metric

LP: Min:e(e)x(e)

st:For every odd cycle C, eCx(e)1

• triangle (metric): i,j,k x(ij)+x(jk)-x(ik)≤ 1
• odd cycle: i1,i2,…,il  j x(ijij+1)≥ 1
• Natural SDP relaxation.
• IG (n).
• Adding triangle + odd cycle cons.:
• IG = ??? (relaxation is stronger than LP).
• Standard random hyperplane rounding [GoemansWilliamson] : ratio = (n½).

x=1 x=-1 x=1

SDP: Min:ij(ij)(1+x(ij))/2

st:X = [x(ij)] is PSD, i x(ii)=1

• triangle (metric): i,j,k x(ij)+x(jk)-x(ik)≤ 1
• odd cycle: i1,i2,…,il  j (1+x(ijij+1))/2 ≥ 1
Concluding remarks
• Part I: Multi-Multiway Cut.
• Ratio that matched Min. Multicut O(log(k)).
• Improve ratio for light instances O(log(Opt)).
• Part II: Min. Uncut.
• Wide open.
• Some naïve techniques don’t work.