On MultiCuts and Related Problems Michael Langberg California Institute of Technology Joint work with Adi Avidor

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

• Part I
• Generalization of both Min. MultiCut and Min.
  Multiway Cut problems.
• Part II
• Minimum Uncut problem.

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Part I Minimum MultiCut

• Input
• G(V,E).
• ? E ? R.
• (si,ti)i1..k.
• Objective
• E ? E that disconnect
• si from ti for all i1..k.
• Measure E of minimum weight.

s1
t3
s3
MultiCut
t2
t1
s2
G(V,E)
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Minimum Multiway Cut

• Input
• G(V,E).
• ? E ? R.
• s1,s2,,sk.
• Objective
• E ? E that disconnect
• si from sj.
• Measure E of minimum weight.

s1
s4
s3
Multiway Cut
s5
s6
s2
G(V,E)
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Multicut vs. Multiway cut.

• Multicut disconnect pairs si,tii1 .. k.
• Multiway Cut disconnect s1,s2,,sk.
• NP-hard, extensively studied in the past.
• Will present known results shortly.
• Roughly
• Multiway Cut lt Multicut.
• Mutiway Cut constant app.
• Multicut only logarithmic app. is known.

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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 i1..k.
• Measure E of minimum weight.

S1
s13
s11
s12
S3
s21
s21
S2
s24
s21
s23
s22
G(V,E)
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Why generalization?

• Input
• G(V,E).
• ? E ? R.
• S1,S2,,Sk Si ? V.
• Multicut ((si,ti)i1..k)
• Each set Sisi,ti.
• Multiway Cut (s1,s2,,sk)
• Singe set S1 of size k.

s13
s11
s12
s21
s21
s24
s21
s23
s22
G(V,E)
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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)i1..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))
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Results and proof techniques

• Multi-Multiway Cut results
• 4ln(k1) approximation.
• 4ln(2OPT) app. (edge weights ? 1).?
• Proof
• Natural LP relaxation.
• Rounding variation of region growing tech.
  LeightonRao, Klein et al., Garg et al.

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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)
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Rounding region growing

• 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.

Multi-Multiway Cut
P1
s13
s11
P2
s12
s21
s21
s24
s21
s23
s22
G(V,E)
Central define the stopping criteria
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Stopping criteria analysis

• Based on that introduced by GargVaziraniYannakaki
  s.
• 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)
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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)
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Min. Uncut previous results

• APX-Hard PapadimitriouYannakakis.
• Min-Uncut lt Min. MultiCut
• KleinRaoAgrawalRavi.
• App. ratio of O(log(V)).
• Remainder of this talk observations on attempt
  to improve app. ratio.

G(V,E)
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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 lt Min. MultiCut

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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 IG not resolved.

x1 x0 x1 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(ijij1)? 1

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What about SDP?

• Natural SDP relaxation.
• IG ??(n).
• Adding triangle odd cycle cons.
• IG ??? (relaxation is stronger than LP).
• Standard random hyperplane rounding
  GoemansWilliamson ratio ?(?n½).

x1 x-1 x1

• SDP Min ?i??j?(ij)(1x(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 (1x(ijij1))/2? 1

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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 dont work.