This talk

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On MultiCuts andOn MultiCuts andRelated Related

ProblemsProblems

Michael LangbergMichael Langberg

California Institute of Technology

Joint work with Adi AvidorJoint work with Adi Avidor

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This talkThis talk

•Part I:Part I:

Generalization of both Generalization of both Min.Min. MultiCutMultiCut and and Min.Min. Multiway CutMultiway Cut problems. problems.

•Part II:Part II:

Minimum UncutMinimum Uncut problem. problem.

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Part I: Minimum MultiCutPart I: Minimum MultiCut

•Input:Input:

•G=(V,E)G=(V,E)..: E : E R R++..

•{(s{(sii,t,tii)})}i=1..ki=1..k..

•Objective:Objective:

•E’ E’ E E that disconnect that disconnect

ssii from from ttii for all for all i=1..ki=1..k..

•Measure:Measure: E’E’ of minimum weight. of minimum weight.

t3

s1

t1s2

t2

s3

G=(V,E)

MultiCut

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Minimum Multiway CutMinimum Multiway Cut

•Input:Input:

•G=(V,E)G=(V,E)..: E : E R R++..

•{s{s11,s,s22,…,s,…,skk}}..



ssii from from ssjj..

•Measure:Measure: E’E’ of minimum of minimum weight.weight.

s4

s1

s6s2

s5

s3

G=(V,E)

Multiway Cut

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Multicut vs. Multiway Multicut vs. Multiway cut.cut.•MulticutMulticut: disconnect pairs : disconnect pairs {s{sii,t,tii}}i=1 .. ki=1 .. k..

•MultiwayMultiway CutCut: disconnect : disconnect {s{s11,s,s22,…,s,…,skk}}..

•NP-hard, extensively studied in the past.NP-hard, extensively studied in the past.

•Will present known results shortly.Will present known results shortly.

•Roughly:Roughly:

•Multiway Cut < Multicut.Multiway Cut < Multicut.

•Mutiway CutMutiway Cut: constant app. : constant app.

•MulticutMulticut: only logarithmic app. is known.: only logarithmic app. is known.

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Our generalization: Our generalization: Minimum Multi-Multiway Minimum Multi-Multiway CutCut•Input:Input:

•G=(V,E)G=(V,E)..: E : E R R++..

•{S{S11,S,S22,…,S,…,Skk}}: : SSii V V..



all vertices in all vertices in SSii for for i=1..ki=1..k..

•Measure:Measure: E’E’ of minimum weight. of minimum weight.

s11

s23

s12

s13

G=(V,E)

S1

s21

s22

s24

S2

s21 s21

S3

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Why generalization?Why generalization?

s11

s23

s12

s13

G=(V,E)

s21

s22

s24

s21 s21

•Input:Input:

•G=(V,E)G=(V,E)..: E : E R R++..

•{S{S11,S,S22,…,S,…,Skk}}: : SSii V V..

•Multicut Multicut ({(s({(sii,t,tii)})}i=1..ki=1..k))

•Each setEach set SSii={s={sii,t,tii}}..

•Multiway Cut:Multiway Cut: ({s({s11,s,s22,…,s,…,skk})})

•Singe set Singe set SS11 of size of size kk..

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Previous resultsPrevious results

MulticutMulticut

MultiwayMultiwayCutCut

Multi-Multi-MultiwayMultiway

CutCut

{(s{(sii,t,tii)})}i=1..ki=1..k

{s{s11,s,s22,…,s,…,skk}}

{S{S11,S,S22,…,S,…,Skk}}

APX-Hard[Dahlhaus et al.]

APX-Hard[Dahlhaus et al.]

APX-Hard[Dahlhaus et al]

O(log(k))[Garg et al.]

O(log(k))

1.34 - k

[Cainescu et al.Karger et al,

CunninghamTang]

“Light inst.”log(Opt)loglog(Opt)[Seymore,Even et al.]

---

“Light inst.”O(log(Opt))

+ Our results

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Results and proof Results and proof techniquestechniques•Multi-Multiway Cut results:Multi-Multiway Cut results:

•4ln(k+1) 4ln(k+1) approximation.approximation.

•4ln(2OPT) 4ln(2OPT) app. (edge weights app. (edge weights 1 1).).

•Proof:Proof:

•Natural LP relaxation.Natural LP relaxation.

•Rounding: variation of Rounding: variation of region growingregion growing tech. tech. [LeightonRao, Klein et al., Garg et al.][LeightonRao, Klein et al., Garg et al.]

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LPLP

LP:LP:

Min:Min: ee(e)x(e)(e)x(e)

st:st:For every path P we For every path P we

want to disconnectwant to disconnect

eePPx(e)x(e)11

x(e)x(e)00

•Correctness: Correctness: x(e)x(e){0,1}{0,1}

s11

s23

s12

s13

G=(V,E)

s21 s22

s24

s21 s21

Multi-Multiway Cut

P

P

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Rounding – region Rounding – region growing growing •From LP: obtained From LP: obtained fractionalfractional

edge values. edge values.

•Implies a semi-metric on Implies a semi-metric on GG..

•SimultaneouslySimultaneously grow balls grow balls around vertices of connected around vertices of connected sets until certain sets until certain criteriacriteria..

•Each ball containes vertices Each ball containes vertices closeclose to center. to center.

•Remove all edges cut by Remove all edges cut by balls.balls.

s11

s23

s12

s13

G=(V,E)

s21 s22

s24

s21 s21

Multi-Multiway Cut

P1

Central: define the stopping criteria

P2

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Stopping criteria + Stopping criteria + analysisanalysis•Based on that introduced by Based on that introduced by

[GargVaziraniYannakakis].[GargVaziraniYannakakis].

•Consider both Consider both volumevolume and and cutcut value of union of balls. value of union of balls.

•Main differences:Main differences:

•SimultaneouslySimultaneously grow grow balls.balls.

•log(Opt)log(Opt): :

•Change Change volumevolume definition. definition.

•Grow Grow largelarge balls only. balls only.

s11

s23

s12

s13

G=(V,E)

s21 s22

s24

s21 s21

Multi-Multiway Cut

P1

P2

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Part II: Minimum UncutPart II: Minimum Uncut

G=(V,E)

Cut

•Input:Input:

•G=(V,E); G=(V,E); : E : E R+. R+.

•Objective: Objective:

•CutCut

•Measure:Measure:

•Minimum weight of Minimum weight of uncutuncut edges (dual to Min. Cut).edges (dual to Min. Cut).

•Find subset Find subset E’E’ of of EE of of minimum weight s.t. minimum weight s.t. G=(V,E-G=(V,E-E’)E’) bipartite. bipartite.

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Min. Uncut: previous Min. Uncut: previous resultsresults

•APX-Hard APX-Hard [PapadimitriouYannakakis][PapadimitriouYannakakis]..

•Min-Uncut < Min. MultiCut Min-Uncut < Min. MultiCut [KleinRaoAgrawalRavi].[KleinRaoAgrawalRavi].

•App. ratio of App. ratio of O(log(|V|))O(log(|V|))..

•Remainder of this talk:Remainder of this talk: observations on attempt to improve observations on attempt to improve app. ratio.app. ratio.

G=(V,E)

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ObservationsObservations

•Our results imply: Our results imply:

O(log(Opt))O(log(Opt)) approximation:approximation:If an undirected graph If an undirected graph GG can be made can be made bipartitebipartite by the deletion of by the deletion of WW edges, then edges, then a set of a set of O(W log W)O(W log W) edges whose deletion edges whose deletion makes the graph bipartite can be makes the graph bipartite can be efficiently found.efficiently found.

•Min-Uncut < Min. MultiCutMin-Uncut < Min. MultiCut

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Observations: LP Observations: LP •Recall:Recall: Min. uncut has ratio Min. uncut has ratio O(log(n))O(log(n))..

•Can show:Can show:

•Natural LP has IG Natural LP has IG (log(n))(log(n))..

•LP enhanced with “triangle” constraints: IG LP enhanced with “triangle” constraints: IG (log(n))(log(n))..

•LP enhanced with “odd cycle” con.: IG LP enhanced with “odd cycle” con.: IG (log(n))(log(n))..

•LP combined with both:LP combined with both: IGIG not resolvednot resolved..

LP: LP: Min:Min: ee(e)x(e)(e)x(e)

st:st: For every odd cycle C, For every odd cycle C, eeCCx(e)x(e)11

•triangle (metric): triangle (metric): i,j,k x(ij)+x(jk)-x(ik)i,j,k x(ij)+x(jk)-x(ik)≤ 1≤ 1

•odd cycle: odd cycle: ii11,i,i22,…,i,…,ill jj x(i x(ijjiij+1j+1))≥ 1≥ 1

x=1 x=0 x=11-x = metric

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What about SDP?What about SDP?•Natural SDP relaxation.Natural SDP relaxation.

•IG IG (n)(n)..

•Adding triangle + odd cycle cons.:Adding triangle + odd cycle cons.:

•IG = ??? (relaxation is stronger than IG = ??? (relaxation is stronger than LP).LP).

•Standard random hyperplane rounding Standard random hyperplane rounding [GoemansWilliamson] [GoemansWilliamson] : ratio = : ratio = ((nn½½))..SDP: SDP: Min:Min: iijj(ij)(1+x(ij))/2(ij)(1+x(ij))/2

st:st: X = [x(ij)] is PSD, X = [x(ij)] is PSD, i x(ii)=1i x(ii)=1

•triangle (metric): triangle (metric): i,j,k x(ij)+x(jk)-x(ik)i,j,k x(ij)+x(jk)-x(ik)≤ 1≤ 1

•odd cycle: odd cycle: ii11,i,i22,…,i,…,ill jj (1+x(i (1+x(ijjiij+1j+1))/2))/2 ≥ 1 ≥ 1

x=1 x=-1 x=1

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Concluding remarksConcluding remarks

•Part IPart I: Multi-Multiway Cut.: Multi-Multiway Cut.

•Ratio that matched Min. Multicut Ratio that matched Min. Multicut O(log(k)).O(log(k)).

•Improve ratio for light instances Improve ratio for light instances O(log(Opt))O(log(Opt))..

•Part IIPart II: Min. Uncut.: Min. Uncut.

•Wide open.Wide open.

•Some naïve techniques don’t work.Some naïve techniques don’t work.

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