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An Improved LP-Based Approximation for Steiner Tree Fabrizio Grandoni Tor Vergata Rome grandoni@disp.uniroma2.it Joint work with J. Byrka, T. Rothvo, L. Sanit` a p. 1/29 The Steiner Tree Problem Def (Steiner tree) Given an undirected

  1. Algorithm IRR • For t = 1 , 2 , . . . ⋄ Compute a (1 + ε ) -apx solution x t for DCR ⋄ Sample a component C = C t with probability C / � p t C := x t D ∈C x t D ⋄ Contract C t and update DCR consequently ⋄ If there is only one terminal, output the sampled components Rem By adding a dummy component in the root, we can assume w.l.o.g. that M := � D ∈C x t D is fixed for all t – p. 12/29

  2. Bridge Lemma – p. 13/29

  3. Bridges Def Given a Steiner tree S and R ′ ⊆ R , the bridges br S,c ( R ′ ) of S w.r.t. R ′ (and edge costs c ) are the edges of S which do not belong to the minimum spanning tree of V ( S ) after the contraction of R ′ S 1 R ′ 8 10 1 2 9 2 1 – p. 14/29

  4. Bridges Def Given a Steiner tree S and R ′ ⊆ R , the bridges br S,c ( R ′ ) of S w.r.t. R ′ (and edge costs c ) are the edges of S which do not belong to the minimum spanning tree of V ( S ) after the contraction of R ′ 0 0 0 0 S 1 R ′ 8 10 1 2 9 2 1 – p. 14/29

  5. Bridges Def Given a Steiner tree S and R ′ ⊆ R , the bridges br S,c ( R ′ ) of S w.r.t. R ′ (and edge costs c ) are the edges of S which do not belong to the minimum spanning tree of V ( S ) after the contraction of R ′ 0 0 0 0 S R ′ 1 2 9 2 1 – p. 14/29

  6. Bridges Def Given a Steiner tree S and R ′ ⊆ R , the bridges br S,c ( R ′ ) of S w.r.t. R ′ (and edge costs c ) are the edges of S which do not belong to the minimum spanning tree of V ( S ) after the contraction of R ′ S 1 R ′ 8 10 br S,c ( R ′ ) 1 2 9 2 1 – p. 14/29

  7. Bridges Def Given a Steiner tree S and R ′ ⊆ R , the bridges br S,c ( R ′ ) of S w.r.t. R ′ (and edge costs c ) are the edges of S which do not belong to the minimum spanning tree of V ( S ) after the contraction of R ′ S 1 R ′ 8 10 br S,c ( R ′ ) 1 2 9 2 1 Rem The most expensive edge on a path between two gray nodes is a bridge – p. 14/29

  8. Bridges Def Given a Steiner tree S and R ′ ⊆ R , the bridges br S,c ( R ′ ) of S w.r.t. R ′ (and edge costs c ) are the edges of S which do not belong to the minimum spanning tree of V ( S ) after the contraction of R ′ S 1 R ′ 8 10 br S,c ( R ′ ) 1 2 9 2 1 Rem Let br S ( R ′ ) = br S,c ( R ′ ) , br S ( R ′ ) := c ( br S ( R ′ )) and br S ( C ) := br S ( R ∩ C ) . – p. 14/29

  9. Bridges Lem For any Steiner tree S on R , br S ( R ) ≥ 1 2 c ( S ) 5 1 2 4 1 3 – p. 15/29

  10. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) – p. 16/29

  11. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 1 8 10 1 2 9 2 1 – p. 16/29

  12. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 1 8 10 1 2 9 2 1 – p. 16/29

  13. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 1 2 9 2 1 – p. 16/29

  14. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 8 1 2 9 2 1 – p. 16/29

  15. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 8 8 1 2 9 2 1 – p. 16/29

  16. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 8 10 1 2 9 2 1 – p. 16/29

  17. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 10 8 10 1 2 9 2 1 – p. 16/29

  18. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 10 8 1 1 2 9 2 1 – p. 16/29

  19. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 10 8 1 1 2 9 2 1 – p. 16/29

  20. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 10 8 1 1 2 9 2 1 – p. 16/29

  21. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • For every C ∈ C , with capacity x C , construct a directed terminal spanning tree Y C on R ∩ C , with capacity x C and edge weights w , as follows 10 8 1 1 2 9 2 1 Rem Y C supports the same flow to the root as C w.r.t. terminals – p. 16/29

  22. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 3 2 – p. 17/29

  23. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 3 2 – p. 17/29

  24. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 3 2 • Replace each component C with the corresponding Y C (cumulating capacities) – p. 17/29

  25. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 4 3 3 2 • Replace each component C with the corresponding Y C (cumulating capacities) – p. 17/29

  26. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 4 4 3 3 2 3 • Replace each component C with the corresponding Y C (cumulating capacities) – p. 17/29

  27. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 4 4 4 2 3 3 2 3 • Replace each component C with the corresponding Y C (cumulating capacities) – p. 17/29

  28. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 4 4 2 3 3 • We obtain a feasible fractional directed terminal spanning tree on a directed graph with V = R and edge costs w – p. 17/29

  29. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 4 4 4 2 3 2 3 3 • We obtain a feasible fractional directed terminal spanning tree on a directed graph with V = R and edge costs w ⇒ By Edmod’s thr there is a cheaper (w.r.t. w ) integral directed terminal spanning tree F – p. 17/29

  30. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 2 3 • We obtain a feasible fractional directed terminal spanning tree on a directed graph with V = R and edge costs w ⇒ By Edmod’s thr there is a cheaper (w.r.t. w ) integral directed terminal spanning tree F – p. 17/29

  31. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) 4 4 3 2 2 3 • The new terminal spanning tree F is more expensive than the original terminal spanning tree T by the cycle-rule – p. 17/29

  32. The Bridge Lemma Lem (Bridge Lemma) For any terminal spanning tree T and any feasible fractional solution x to DCR, � C ∈C x C · br T ( C ) ≥ c ( T ) • Summarizing � � x C · br T ( C ) = x C · w ( Y C ) ≥ w ( F ) ≥ c ( T ) � �� � C ∈C C ∈C w -cost of � �� � integral w -cost of terminal fractional spanning tree terminal spanning tree – p. 18/29

  33. Approximation Factor – p. 19/29

  34. A First Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 + ln 2 + ε ) opt f – p. 20/29

  35. A First Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 + ln 2 + ε ) opt f Cor The integrality gap of DCR is at most 1 + ln 2 < 1 . 7 – p. 20/29

  36. A First Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 + ln 2 + ε ) opt f x t M c ( C )] ≤ 1 + ε X X X X E [ c ( C t )] ≤ C E [ opt f,t ] E [ apx ] = E [ M C t ≥ 1 t ≥ 1 t ≥ 1 M ln 2 ≤ 1 + ε opt f + 1 + ε X X E [ c ( T t )] M M t =1 t>M ln 2 • T t is a minimum terminal spanning tree at step t – p. 21/29

  37. A First Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 + ln 2 + ε ) opt f x t M c ( C )] ≤ 1 + ε X X X X E [ c ( C t )] ≤ C E [ opt f,t ] E [ apx ] = E [ M C t ≥ 1 t ≥ 1 t ≥ 1 M ln 2 ≤ 1 + ε opt f + 1 + ε X X E [ c ( T t )] M M t =1 t>M ln 2 Lem For any t , E [ c ( T t +1 )] ≤ (1 − 1 M ) c ( T t ) x t E [ c ( T t +1 )] ≤ c ( T t ) − E [ br T t ( C t )] = c ( T t ) − X C M br T t ( C ) C Bridge Lem c ( T t ) − 1 M c ( T t ) ≤ – p. 21/29

  38. A First Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 + ln 2 + ε ) opt f x t M c ( C )] ≤ 1 + ε X X X X E [ c ( C t )] ≤ C E [ opt f,t ] E [ apx ] = E [ M C t ≥ 1 t ≥ 1 t ≥ 1 M ln 2 ≤ 1 + ε opt f + 1 + ε X X E [ c ( T t )] M M t =1 t>M ln 2 Lem For any t , E [ c ( T t +1 )] ≤ (1 − 1 M ) c ( T t ) x t E [ c ( T t +1 )] ≤ c ( T t ) − E [ br T t ( C t )] = c ( T t ) − X C M br T t ( C ) C Bridge Lem c ( T t ) − 1 M c ( T t ) ≤ Cor E [ c ( T t )] ≤ (1 − 1 M ) t − 1 c ( T 1 ) ≤ (1 − 1 M ) t − 1 2 opt f – p. 21/29

  39. A First Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 + ln 2 + ε ) opt f x t M c ( C )] ≤ 1 + ε X X X X E [ c ( C t )] ≤ C E [ opt f,t ] E [ apx ] = E [ M C t ≥ 1 t ≥ 1 t ≥ 1 M ln 2 ≤ 1 + ε opt f + 1 + ε X X E [ c ( T t )] M M t =1 t>M ln 2 « t − 1 „ 1 1 − 1 ≤ opt f (1 + ε ) ln 2 + 2 opt f (1 + ε ) X M M t>M ln 2 ≤ (1 + ε )(ln 2 + 2 e − ln 2 ) · opt f – p. 21/29

  40. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt – p. 22/29

  41. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Rem This bound might not hold w.r.t. opt f – p. 22/29

  42. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt x t M c ( C )] ≤ 1 + ε X E [ c ( C t )] ≤ X X X E [ opt f,t ] C E [ apx ] = E [ M t ≥ 1 t ≥ 1 C t ≥ 1 M ln 4 ≤ 1 + ε E [ c ( S t )] + 1 + ε X X E [ c ( T t )] M M t =1 t> M ln 4 • S t is a minimum Steiner tree at step t – p. 22/29

  43. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) – p. 22/29

  44. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) • Construct a terminal spanning tree ( Y t , w ) w.r.t. S t and all its terminals R t = R ∩ S t as in the proof of the bridge lemma. – p. 22/29

  45. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) • Construct a terminal spanning tree ( Y t , w ) w.r.t. S t and all its terminals R t = R ∩ S t as in the proof of the bridge lemma. • Let b ( e ) ∈ S t be the bridge associated to e ∈ Y t . – p. 22/29

  46. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) • Construct a terminal spanning tree ( Y t , w ) w.r.t. S t and all its terminals R t = R ∩ S t as in the proof of the bridge lemma. • Let b ( e ) ∈ S t be the bridge associated to e ∈ Y t . b ( e 2 ) 4 e 2 4 b ( e 1 ) e 1 3 2 3 – p. 22/29

  47. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) • S ′ := S t - { b ( e ) ∈ S t | e ∈ br Y t , w ( C t ) } is a feasible Steiner tree at step t + 1 b ( e 2 ) 4 e 2 4 b ( e 1 ) e 1 3 2 3 – p. 23/29

  48. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) • S ′ := S t - { b ( e ) ∈ S t | e ∈ br Y t , w ( C t ) } is a feasible Steiner tree at step t + 1 b ( e 2 ) 4 e 2 4 b ( e 1 ) e 1 3 2 3 – p. 23/29

  49. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) • S ′ := S t - { b ( e ) ∈ S t | e ∈ br Y t , w ( C t ) } is a feasible Steiner tree at step t + 1 b ( e 2 ) 4 e 2 4 b ( e 1 ) 3 2 – p. 23/29

  50. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) • S ′ := S t - { b ( e ) ∈ S t | e ∈ br Y t , w ( C t ) } is a feasible Steiner tree at step t + 1 b ( e 2 ) 4 e 2 4 2 – p. 23/29

  51. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) E [ c ( S t +1 )] ≤ E [ c ( S ′ )] = c ( S t ) − E [ c ( { b ( e ) ∈ S t | e ∈ br Y t , w ( C t ) } )] = c ( S t ) − E [ br Y t , w ( C t )] = c ( S t ) − 1 X x t C br Y t , w ( C ) M C c ( S t ) − 1 Bridge Lem M w ( Y t ) ≤ = c ( S t ) − 1 M br S t , c ( R t ) c ( S t ) ≤ c ( S t ) − 1 2 M – p. 24/29

  52. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt Lem For any t , E [ c ( S t +1 )] ≤ (1 − 1 2 M ) c ( S t ) E [ c ( S t +1 )] ≤ E [ c ( S ′ )] = c ( S t ) − E [ c ( { b ( e ) ∈ S t | e ∈ br Y t , w ( C t ) } )] = c ( S t ) − E [ br Y t , w ( C t )] = c ( S t ) − 1 X x t C br Y t , w ( C ) M C c ( S t ) − 1 Bridge Lem M w ( Y t ) ≤ = c ( S t ) − 1 M br S t , c ( R t ) c ( S t ) ≤ c ( S t ) − 1 2 M Cor E [ c ( S t )] ≤ (1 − 1 1 2 M ) t − 1 c ( S 1 ) = (1 − 2 M ) t − 1 opt – p. 24/29

  53. A Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (1 . 5 + ε ) opt x t j )] ≤ 1 + ε j X E [ c ( C t )] ≤ X X M c ( C t X E [ opt f,t ] E [ apx ] = E [ M j t ≥ 1 t ≥ 1 t ≥ 1 M ln 4 ≤ 1 + ε E [ c ( S t )] + 1 + ε X X E [ c ( T t )] M M t =1 t>M ln 4 M ln 4 ≤ (1 + ε 1 2(1 − 1 2 M ) t − 1 + X X M ) t − 1 ) opt ) · ( (1 − M t =1 t>M ln 4 ≤ (1 + ε )(2 − 2 e − ln(4) / 2 + 2 e − ln(4) ) · opt – p. 25/29

  54. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt – p. 26/29

  55. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 • We define a random terminal spanning tree W ( witness tree) – p. 26/29

  56. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 0 0 0 0 • We define a random terminal spanning tree W ( witness tree) – p. 26/29

  57. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 • We define a random terminal spanning tree W ( witness tree) – p. 26/29

  58. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 • We associate to each e in the Steiner tree S the edges W ( e ) of W such that the corresponding path in S contains e • Observe that | W ( e ) | is 1 , 2 . . . with probability 1 2 , 1 4 , . . . – p. 26/29

  59. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 • We associate to each e in the Steiner tree S the edges W ( e ) of W such that the corresponding path in S contains e • Observe that | W ( e ) | is 1 , 2 . . . with probability 1 2 , 1 4 , . . . – p. 26/29

  60. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 • For any sampled component C t , we delete from W a random set of bridges such that each edge of W is deleted with probability ≥ 1 /M ( ⇐ Farkas’ lemma+Bridge lemma) • When W ( e ) is deleted, we delete e from S – p. 26/29

  61. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 • For any sampled component C t , we delete from W a random set of bridges such that each edge of W is deleted with probability ≥ 1 /M ( ⇐ Farkas’ lemma+Bridge lemma) • When W ( e ) is deleted, we delete e from S – p. 26/29

  62. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 • For any sampled component C t , we delete from W a random set of bridges such that each edge of W is deleted with probability ≥ 1 /M ( ⇐ Farkas’ lemma+Bridge lemma) • When W ( e ) is deleted, we delete e from S – p. 26/29

  63. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 • For any sampled component C t , we delete from W a random set of bridges such that each edge of W is deleted with probability ≥ 1 /M ( ⇐ Farkas’ lemma+Bridge lemma) • When W ( e ) is deleted, we delete e from S – p. 26/29

  64. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 4 3 1 • For any sampled component C t , we delete from W a random set of bridges such that each edge of W is deleted with probability ≥ 1 /M ( ⇐ Farkas’ lemma+Bridge lemma) • When W ( e ) is deleted, we delete e from S – p. 26/29

  65. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 3 1 • For any sampled component C t , we delete from W a random set of bridges such that each edge of W is deleted with probability ≥ 1 /M ( ⇐ Farkas’ lemma+Bridge lemma) • When W ( e ) is deleted, we delete e from S – p. 26/29

  66. An Even Better Bound Thr Algorithm IRR computes a solution of expected cost ≤ (ln 4 + ε ) opt 1 5 2 3 1 • Each e ∈ S survives in expectation M · ln 4 rounds – p. 26/29

  67. Derandomization Thr There is a ln 4 + ε deterministic approximation algorithm for Steiner tree – p. 27/29

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