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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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slide1

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

this talk
This talk
  • Part I:

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

  • Part II:

Minimum Uncut problem.

part i minimum multicut
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
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 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
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
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
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
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.]
slide10
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
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
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
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: previous results
  • APX-Hard [PapadimitriouYannakakis].
  • 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
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
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
what about sdp
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).

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
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.