An Approximation Algorithm for Requirement cut on graphs

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• An Approximation Algorithm for Requirement cut on
  graphs
• Viswanath Nagarajan
• Joint work with R. Ravi

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Cut problems

• Undirected G(V,E)
• Edge costs c E ? R
• Remove minimum cost set of edges satisfying some
  requirement

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Example Multicut

• Pairs (s1,t1), (s2,t2), , (sk,tk)
• Separate each pair
• O(log k) approximation GVY93

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Example Steiner multicut

• Groups X1, X2, , Xg µ V
• Separate each group
• O(log3 (gt)) approximation KPRT97
• (t maxi Xi)

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More examples

• k-cut 2 SV 95
• Multiway cut 1.34 KKSTY 99
• Steiner k-cut 2 CGN 03
• Multi-multiway cut log k AL04

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Requirement cut

• Groups X1, X2, , Xg µ V
• Requirements ri (0 ri Xi)
• Separate group Xi into ri pieces
• Generalizes previous cut problems

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Containment of cut problems
Requirement cut
Steiner multicut
Steiner k-cut
k-cut
Multi-multiway cut
Multicut
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Our results

• O(log n log(gR)) approximation
• n number of vertices
• g number of groups
• R maximum requirement
• O(log(gR)) approximation on trees

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IP formulation

• min ?e2 E ce de
• s.t.
• ?e2 Ti de ri-1 8 Ti Steiner tree on Xi,
  8 i1, , g
• de 2 0,1 8 e2 E

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Solving LP relaxation

• Minimum Steiner tree NP hard!
• Relax to spanning trees
• Require that d is metric

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LP relaxation

• min ?e2 E ce de
• s.t.
• ?e2 Ti de ri 1 8 Ti spanning tree on Xi
• 8 i 1, , g
• d metric
• de2 0,1 8 e2 E

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Requirement cut on trees

• Input graph G is a tree
• Reduction from set cover ) ?(log g) hard
• LP rounding yields O(log(gR)) approximation

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Tree rounding

• Solve LP to get d (OPTlp ?e2 E ce de)
• Define de min2 de, 1
• Repeat O(log(gR)) times
• Pick each edge of tree G with probability de

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Tree rounding

• Theorem Randomized rounding for requirement cut
  on trees yields a solution of cost at most
  O(log(gR)) OPTlp
• Cost in single phase 2 OPTlp
• Total cost of rounding O(log (gR)) OPTlp
• Need to satisfy all requirements

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Single phase

• ci current number of components of Xi
• Residual requirement of Xi ri ci
• Lemma In each phase of randomized rounding, the
  total residual requirement reduces by a factor of
  4/3, in expectation.

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Bounding number of phases

• Initial requirement gR
• Expected requirement after phase k,
  ERk (¾)k gR
• k 4ln(gR) ) ERk ½

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Rounding in single phase

• Current forest F
• Fi forest induced by Xi
• Hi shortcut Fi over degree 2 Steiner vertices

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Single phase - analysis

• Removing edges of Hi new components in Fi
• Lemma Expected number of edges of Hi removed
  is at least (1-1/e) d(Hi)
• Here, d(Hi) ?e2 Hi de

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Single phase - analysis

• Claim In any Steiner forest Hi with each non
  terminal having degree 3, removing any m edges
  results in at least d(m1)/2e new terminal
  components
• Expected number of new components containing
  Xi (1- 1/e) ½ d(Hi) ¼ d(Hi)

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Single phase - analysis

• Extend Hi to Steiner tree on Xi
• Add ci 1 edges
• d(Hi) ci 1 ri 1 ( Sp. tree constraint)
• d(Hi) ri ci residual requirement of Xi

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Single phase - analysis

• Number of new components ¼ (ri-ci)
• new requirement of group Xi
• old requirement number of new components
• ¾ old requirement
• total new requirement
• ¾ total old requirement (All in
  expectation)

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General graphs

• Solve LP on G to get metric d
• Use FRT embedding to tree metric (?,T)
• (edges of T cuts in G)
• Use tree rounding on T
• O(log n log(gR)) approximation

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Conclusions

• Introduced requirement cut problem
• O(log n log(gR)) integrality gap
• Improvement for planar graphs?
• Even on trees, gap between ?(log g) lower bound,
  O(log(gR)) upper bound

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Thank you!
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Extra claim

• Claim Minimum Steiner tree on Xi w.r.t. d is
  at least ri 1

d costs on edges
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Single phase - analysis

• Lemma Expected number of edges of Hi removed
  is at least (1-1/e) d(Hi)
• (u,v) edge of Hi
• P path connecting u and v in Fi

Pµ Fi
u
v
(u,v)2 Hi
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Lemma contd.

• Pr separating u and v 1 - ?e2 P (1-de)
• 1 e-d(P)
• A) d(P) 1
• Pr 1 e-d(P) (1-1/e) d(P) (1-1/e)
  du,v
• B) d(P) 1
• Pr 1 e-d(P) 1 1/e (1-1/e) du,v