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

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

  3. 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)

  4. 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)

  5. 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.

  6. 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)

  7. 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)

  8. 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))

  9. 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.]

  10. 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)

  11. 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

  12. 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)

  13. 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)

  14. 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)

  15. 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

  16. 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

  17. 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

  18. 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.

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