Separators with Non-Hereditary Properties

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Separators with Non-Hereditary Properties

  1. 1. separators
  2. 2. separatorswith non-hereditary properties
  3. 3. separatorswith non-hereditary properties Pinar Heggernes, Pim van’t Hof, Dániel Marx, and Yngve Villanger
  4. 4. OJ`çååÉÅíÉÇ=pÉé~ê~íçêë
  5. 5. OJ`çååÉÅíÉÇ=pÉé~ê~íçêë The Treewidth Reduction Theorem
  6. 6. OJ`çååÉÅíÉÇ=pÉé~ê~íçêë The Treewidth Reduction TheoremOJ`çååÉÅíÉÇ=píÉáåÉê=qêÉÉë
  7. 7. OJ`çååÉÅíÉÇ=pÉé~ê~íçêë The Treewidth Reduction TheoremOJ`çååÉÅíÉÇ=píÉáåÉê=qêÉÉë Some Structural Observations
  8. 8. PRELIMINARIES
  9. 9. Cliques
  10. 10. Polynomial Time Cliques
  11. 11. Independent  Set
  12. 12. Fixed Parameter Tractable Independent  Set
  13. 13. Fixed Parameter TractableYour  favorite  Hereditary  Property
  14. 14. Fixed Parameter Tractable Tarjan; Marx, Sullivan and RazgonYour  favorite  Hereditary  Property
  15. 15. Tarjan; Marx, Sullivan and RazgonWhat  about  non-­hereditary  properties?
  16. 16. ? Tarjan; Marx, Sullivan and RazgonWhat  about  non-­hereditary  properties?
  17. 17. Connected Separators
  18. 18. 2- Connected Separators
  19. 19. c - Connected Separators
  20. 20. c - Connected Separators 0- Regular Separators
  21. 21. c - Connected Separators c -Regular Separators
  22. 22. c - Connected Separators c -Regular Separators1-Diameter Separators
  23. 23. c - Connected Separators c -Regular Separatorsc -Diameter Separators
  24. 24. 2-CONNECTED SEPARATORS
  25. 25. The 2-connected Separator Problem
  26. 26. The 2-connected Separator Problem
  27. 27. The 2-connected Separator Problem A  (s,t)  separator  of  size  at  most  kthat  induces  a  2-­connected  subgraph.
  28. 28. The 2-connected Separator Problem“Easy”  on  graphs  of  small  treewidth. (Due  to  MSO  expressibility.)
  29. 29. The 2-connected Separator Problem“Easy”  on  graphs  of  small  treewidth. (Due  to  MSO  expressibility.) General  graphs Equivalent  instances  with  small   treewidth
  30. 30. The 2-connected Separator Problem The  Treewidth  Reduction   Theorem General  graphs Equivalent  instances  with  small   treewidth
  31. 31. The 2-connected Separator Problem
  32. 32. The 2-connected Separator Problem
  33. 33. The 2-connected Separator ProblemH  contains  all  minimal  (s,t)  separators  and  tw(H)  =  g(k)
  34. 34. The 2-connected Separator ProblemH  contains  all  minimal  (s,t)  separators  and  tw(H)  =  g(k)
  35. 35. The 2-connected Separator ProblemH  contains  all  minimal  (s,t)  separators  and  tw(H)  =  g(k)
  36. 36. The 2-connected Separator ProblemH  contains  all  minimal  (s,t)  separators  and  tw(H)  =  g(k)
  37. 37. The 2-connected Separator ProblemH  contains  all  minimal  (s,t)  separators  and  tw(H)  =  g(k)
  38. 38. The 2-connected Separator ProblemH  contains  all  minimal  (s,t)  separators  and  tw(H)  =  g(k)
  39. 39. The 2-connected Separator Problem G{H}  =  torso  of  H  in  GH  contains  all  minimal  (s,t)  separators  and  tw(H)  =  g(k)
  40. 40. The 2-connected Separator Problem G{H}  =  torso  of  H  in  G 2-­connectedH  contains  all  minimal  (s,t)  separators  and  tw(H)  =  g(k)
  41. 41. The 2-connected Separator Problem G{H}  =  torso  of  H  in  G witnesses 2-­connectedH  contains  all  minimal  (s,t)  separators  and  tw(H)  =  g(k)
  42. 42. k k2q · g(k) · 2
  43. 43. k k2q · g(k) · 2
  44. 44. k k2q · g(k) · 2
  45. 45. k k2q · g(k) · 2
  46. 46. 2-­connected(s,t)  separator
  47. 47. 2-­connected by construction(s,t)  separator
  48. 48. 2-­connected by construction(s,t)  separator contains a minimal separator
  49. 49. ` k k2q · g(k) · 2
  50. 50. ` k k2q · g(k) · 2
  51. 51. 2-CONNECTED STEINER TREE
  52. 52. The 2-connected Steiner Tree Problem Some structural Discoveries An Algorithm
  53. 53. The 2-connected Steiner Tree Problem
  54. 54. The 2-connected Steiner Tree Problem
  55. 55. The 2-connected Steiner Tree Problem
  56. 56. The 2-connected Steiner Tree Problem terminals
  57. 57. The 2-connected Steiner Tree Problem terminals
  58. 58. The 2-connected Steiner Tree Problem terminals a 2-connected subgraph
  59. 59. The 2-connected Steiner Tree Problem terminals the terminals are 2-connected
  60. 60. The 2-connected Steiner Tree Problem terminals the terminals are 2-connected
  61. 61. The 2-connected Steiner Tree Problem terminals the terminals are 2-connected
  62. 62. The 2-connected Steiner Tree Problem terminals the terminals are 2-connected
  63. 63. The 2-connected Steiner Tree Problem terminals the terminals are 2-connected the subgraph (if minimal) is 2-connected
  64. 64. The 2-connected Steiner Tree Problem terminals
  65. 65. The 2-connected Steiner Tree Problem terminals the terminals are c-connected the subgraph (if minimal) is c-connected
  66. 66. The 2-connected Steiner Tree Problem terminals the terminals are c-connected the subgraph (if minimal) is c-connected
  67. 67. The 2-connected Steiner Tree Problem terminals the terminals are c-connected the subgraph (if minimal) is c-connected
  68. 68. The 2-connected Steiner Tree Problem terminals the terminals are c-connected the subgraph (if minimal) is c-connected
  69. 69. The 2-connected Steiner Tree Problem terminals the terminals are c-connected the subgraph (if minimal) is c-connected
  70. 70. The 2-connected Steiner Tree Problem terminals
  71. 71. The 2-connected Steiner Tree Problem terminals
  72. 72. The 2-connected Steiner Tree Problem Claim:  HT    is  a  forest.
  73. 73. The 2-connected Steiner Tree Problem Claim:  HT    is  a  forest.
  74. 74. The 2-connected Steiner Tree Problem Case 1: When C does not separate T. Claim:  HT    is  a  forest.
  75. 75. The 2-connected Steiner Tree Problem Case 1: When C does not separate T. Claim:  HT    is  a  forest.
  76. 76. The 2-connected Steiner Tree Problem Case 2: When C separates T. Claim:  HT    is  a  forest.
  77. 77. The 2-connected Steiner Tree Problem Case 2: When C separates T. Claim:  HT    is  a  forest.
  78. 78. The 2-connected Steiner Tree Problem Case 2: When C separates T. Claim:  HT    is  a  forest.
  79. 79. The 2-connected Steiner Tree Problem Case 2: When C separates T. Claim:  HT    is  a  forest.
  80. 80. The 2-connected Steiner Tree Problem Case 2: When C separates T. Claim:  HT    is  a  forest.
  81. 81. The 2-connected Steiner Tree Problem Case 2: When C separates T. Claim:  HT    is  a  forest.
  82. 82. The 2-connected Steiner Tree Problem Case 2: When C separates T. Claim:  HT    is  a  forest.
  83. 83. The 2-connected Steiner Tree Problem Claim:  HT    is  a  forest.
  84. 84. The 2-connected Steiner Tree Problem
  85. 85. The 2-connected Steiner Tree Problem
  86. 86. The 2-connected Steiner Tree Problem
  87. 87. The 2-connected Steiner Tree Problem Guess the structure of H.
  88. 88. The 2-connected Steiner Tree Problem Try  all  graphs  H  for   which  HT  is  a  forest.
  89. 89. The 2-connected Steiner Tree Problem Try  all  graphs  H  for   which  HT  is  a  forest.
  90. 90. The 2-connected Steiner Tree Problem Try  all  graphs  H  for   which  HT  is  a  forest. Map  H  into  G  such  that  vertices  in  H   are  assigned  to  their  “candidates”  in  G.
  91. 91. The 2-connected Steiner Tree Problem Try  all  graphs  H  for   which  HT  is  a  forest. Map  H  into  G  such  that  vertices  in  H   are  assigned  to  their  “candidates”  in  G.
  92. 92. H
  93. 93. H G
  94. 94. H G
  95. 95. H G
  96. 96. H G
  97. 97. H Prune  the  colorclasses  to  get  rid  of non-­candidates. G
  98. 98. H Delete  irrelevant  edges. G
  99. 99. H Performa  breadth-­first   search. G

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