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Convex geometric graphs Complexity st path augmentation Augmenting the Connectivity of Planar and Geometric Graphs Ignaz Rutter Alexander Wolff Universitt Karlsruhe TU Eindhoven Ignaz Rutter and Alexander Wolff 1 24 Connectivity

  1. Convex geometric graphs Complexity s–t path augmentation Convex geometric graphs Theorem Geometric PVCA and geometric PECA can be solved in linear time for connected convex geometric graphs. PECA: Ignaz Rutter and Alexander Wolff 8 24 Connectivity Augmentation

  2. Convex geometric graphs Complexity s–t path augmentation Convex geometric graphs Theorem Geometric PVCA and geometric PECA can be solved in linear time for connected convex geometric graphs. PECA: Ignaz Rutter and Alexander Wolff 8 24 Connectivity Augmentation

  3. Convex geometric graphs Complexity s–t path augmentation Convex geometric graphs Theorem Geometric PVCA and geometric PECA can be solved in linear time for connected convex geometric graphs. PECA: Ignaz Rutter and Alexander Wolff 8 24 Connectivity Augmentation

  4. Convex geometric graphs Complexity s–t path augmentation Convex geometric graphs Theorem Geometric PVCA and geometric PECA can be solved in linear time for connected convex geometric graphs. PECA: Ignaz Rutter and Alexander Wolff 8 24 Connectivity Augmentation

  5. Convex geometric graphs Complexity s–t path augmentation Convex geometric graphs Theorem Geometric PVCA and geometric PECA can be solved in linear time for connected convex geometric graphs. PECA: Ignaz Rutter and Alexander Wolff 8 24 Connectivity Augmentation

  6. Convex geometric graphs Complexity s–t path augmentation Overview Convex geometric graphs 1 Complexity 2 s–t path augmentation 3 Ignaz Rutter and Alexander Wolff 9 24 Connectivity Augmentation

  7. Convex geometric graphs Complexity s–t path augmentation Complexity of PECA Theorem PECA is NP-hard. Ignaz Rutter and Alexander Wolff 10 24 Connectivity Augmentation

  8. Convex geometric graphs Complexity s–t path augmentation Complexity of PECA Theorem PECA is NP-hard. Proof: gadget proof reduction from P LANAR 3S AT c 4 c 2 c 1 c 3 x 2 x 4 x 5 x n x 1 x 3 . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 10 24 Connectivity Augmentation

  9. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  10. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  11. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  12. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  13. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  14. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  15. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  16. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  17. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  18. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  19. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  20. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  21. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  22. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  23. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  24. Convex geometric graphs Complexity s–t path augmentation Variable c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  25. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  26. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  27. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  28. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  29. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  30. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  31. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  32. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  33. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  34. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  35. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  36. Convex geometric graphs Complexity s–t path augmentation Pipe c 4 attach clause c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 attach variable Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  37. Convex geometric graphs Complexity s–t path augmentation Literals c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . . . . c 5 c 6 c 7 Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  38. Convex geometric graphs Complexity s–t path augmentation Clause c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 pipe pipe attach pipe Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  39. Convex geometric graphs Complexity s–t path augmentation Clause c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 pipe pipe attach pipe Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  40. Convex geometric graphs Complexity s–t path augmentation Clause c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 pipe pipe attach pipe Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  41. Convex geometric graphs Complexity s–t path augmentation Clause c 4 c 2 c 1 c 3 x 1 x 2 x 3 x 4 x 5 x n . . . c 5 c 6 c 7 pipe pipe attach pipe Ignaz Rutter and Alexander Wolff 11 24 Connectivity Augmentation

  42. Convex geometric graphs Complexity s–t path augmentation Complexity of geometric PVCA / geometric PECA We conclude: Theorem PECA is NP-hard. Ignaz Rutter and Alexander Wolff 12 24 Connectivity Augmentation

  43. Convex geometric graphs Complexity s–t path augmentation Complexity of geometric PVCA / geometric PECA We conclude: Theorem PECA is NP-hard. Theorem Geometric PVCA and geometric PECA are NP-complete. Ignaz Rutter and Alexander Wolff 12 24 Connectivity Augmentation

  44. Convex geometric graphs Complexity s–t path augmentation Complexity of geometric PVCA / geometric PECA We conclude: Theorem PECA is NP-hard. Theorem Geometric PVCA and geometric PECA are NP-complete. yet another gadget proof ;-) Ignaz Rutter and Alexander Wolff 12 24 Connectivity Augmentation

  45. Convex geometric graphs Complexity s–t path augmentation Trees Theorem Geometric PVCA and geometric PECA are NP-complete. Ignaz Rutter and Alexander Wolff 13 24 Connectivity Augmentation

  46. Convex geometric graphs Complexity s–t path augmentation Trees Theorem Geometric PVCA and geometric PECA are NP-complete. Corollary ... even in the case of trees. Ignaz Rutter and Alexander Wolff 13 24 Connectivity Augmentation

  47. Convex geometric graphs Complexity s–t path augmentation Trees Theorem Geometric PVCA and geometric PECA are NP-complete. Corollary ... even in the case of trees. Proof: Ignaz Rutter and Alexander Wolff 13 24 Connectivity Augmentation

  48. Convex geometric graphs Complexity s–t path augmentation Trees Theorem Geometric PVCA and geometric PECA are NP-complete. Corollary ... even in the case of trees. Proof: Ignaz Rutter and Alexander Wolff 13 24 Connectivity Augmentation

  49. Convex geometric graphs Complexity s–t path augmentation Overview Convex geometric graphs 1 Complexity 2 s–t path augmentation 3 Ignaz Rutter and Alexander Wolff 14 24 Connectivity Augmentation

  50. Convex geometric graphs Complexity s–t path augmentation Geometric path augmentation Problem: s – t k -C ONN A UG connected plane geometric graph G = ( V , E ) Given: and two vertices s and t in G . Find: Minimal set of vertex pairs E ′ , such that G ′ = ( V , E ∪ E ′ ) is plane and G ′ contains k edge-disjoint s – t paths. Ignaz Rutter and Alexander Wolff 15 24 Connectivity Augmentation

  51. Convex geometric graphs Complexity s–t path augmentation Geometric path augmentation Problem: s – t k -C ONN A UG connected plane geometric graph G = ( V , E ) Given: and two vertices s and t in G . Find: Minimal set of vertex pairs E ′ , such that G ′ = ( V , E ∪ E ′ ) is plane and G ′ contains k edge-disjoint s – t paths. example for k = 2: s t Ignaz Rutter and Alexander Wolff 15 24 Connectivity Augmentation

  52. Convex geometric graphs Complexity s–t path augmentation Geometric path augmentation Problem: s – t k -C ONN A UG connected plane geometric graph G = ( V , E ) Given: and two vertices s and t in G . Find: Minimal set of vertex pairs E ′ , such that G ′ = ( V , E ∪ E ′ ) is plane and G ′ contains k edge-disjoint s – t paths. example for k = 2: s t Ignaz Rutter and Alexander Wolff 15 24 Connectivity Augmentation

  53. Convex geometric graphs Complexity s–t path augmentation Worst-Case analysis for s – t 2-Aug Theorem G = ( V , E ) a plane connected geometric graph, s , t ∈ V, n = | V | . Then G has an s–t 2-Aug of size n / 2 . 1 Such an s–t 2-Aug can be computed in O ( n ) time. 2 Ignaz Rutter and Alexander Wolff 16 24 Connectivity Augmentation

  54. Convex geometric graphs Complexity s–t path augmentation Worst-Case analysis for s – t 2-Aug Theorem G = ( V , E ) a plane connected geometric graph, s , t ∈ V, n = | V | . Then G has an s–t 2-Aug of size n / 2 . 1 Such an s–t 2-Aug can be computed in O ( n ) time. 2 Proof: Compute any triangulation T of G . 1 Find an s – t path π with | π | ≤ n / 2 in T . 2 Compute an augmentation from π with ≤ | π | edges. 3 Ignaz Rutter and Alexander Wolff 16 24 Connectivity Augmentation

  55. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . s t Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  56. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . N 1 ( t ) N 1 ( s ) s t Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  57. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . N 2 ( s ) N 2 ( t ) s t Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  58. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . N 3 ( s ) N 3 ( t ) s t Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  59. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . N 4 ( t ) N 4 ( s ) s t Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  60. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . N 4 ( t ) N 4 ( s ) s t π Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  61. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . N 4 ( t ) N 4 ( s ) s t π N k ( s ) ∩ N k ( t ) � = ∅ ⇒ ∃ s – t path π with | π | ≤ 2 k . Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  62. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . N 4 ( t ) N 4 ( s ) s t π N k ( s ) ∩ N k ( t ) � = ∅ ⇒ ∃ s – t path π with | π | ≤ 2 k . | N i ( v ) | ≥ 2 i + 1 for every vertex v of a triangulation. Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  63. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . N 4 ( t ) N 4 ( s ) s t π N k ( s ) ∩ N k ( t ) � = ∅ ⇒ ∃ s – t path π with | π | ≤ 2 k . | N i ( v ) | ≥ 2 i + 1 for every vertex v of a triangulation. If N i ( s ) ∩ N i ( t ) = ∅ then | N i ( s ) ∪ N i ( t ) | ≥ 2 + 4 i . ⇒ Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  64. Convex geometric graphs Complexity s–t path augmentation Triangulation contains s – t path of length ≤ n / 2. Consider growing neighborhoods N i ( s ) , N i ( t ) . N 4 ( t ) N 4 ( s ) s t π N k ( s ) ∩ N k ( t ) � = ∅ ⇒ ∃ s – t path π with | π | ≤ 2 k . | N i ( v ) | ≥ 2 i + 1 for every vertex v of a triangulation. If N i ( s ) ∩ N i ( t ) = ∅ then | N i ( s ) ∪ N i ( t ) | ≥ 2 + 4 i . ⇒ ⇒ after k � n / 4 steps the whole graph is covered by N k ( s ) ∪ N k ( t ) . Ignaz Rutter and Alexander Wolff 17 24 Connectivity Augmentation

  65. Convex geometric graphs Complexity s–t path augmentation How to construct an augmentation from π Consider each edge e of π e is not in G ⇒ add e to G . 1 Ignaz Rutter and Alexander Wolff 18 24 Connectivity Augmentation

  66. Convex geometric graphs Complexity s–t path augmentation How to construct an augmentation from π Consider each edge e of π e is not in G ⇒ add e to G . 1 ⇒ e belongs to G , e is no bridge nothing to do 2 Ignaz Rutter and Alexander Wolff 18 24 Connectivity Augmentation

  67. Convex geometric graphs Complexity s–t path augmentation How to construct an augmentation from π Consider each edge e of π e is not in G ⇒ add e to G . 1 ⇒ e belongs to G , e is no bridge nothing to do 2 e belongs to G , e is a bridge ... 3 e Ignaz Rutter and Alexander Wolff 18 24 Connectivity Augmentation

  68. Convex geometric graphs Complexity s–t path augmentation How to construct an augmentation from π Consider each edge e of π e is not in G ⇒ add e to G . 1 ⇒ e belongs to G , e is no bridge nothing to do 2 e belongs to G , e is a bridge ... 3 H 2 H 1 e t s Ignaz Rutter and Alexander Wolff 18 24 Connectivity Augmentation

  69. Convex geometric graphs Complexity s–t path augmentation How to construct an augmentation from π Consider each edge e of π e is not in G ⇒ add e to G . 1 ⇒ e belongs to G , e is no bridge nothing to do 2 e belongs to G , e is a bridge ... 3 H 2 H 1 e t s Ignaz Rutter and Alexander Wolff 18 24 Connectivity Augmentation

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