A Kernel for Planar F-deletion: The Connected Case

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A Kernel for Planar F-deletion: The Connected Case

  1. 1. Polynomial Kernels for Planar F-deletion
  2. 2. Polynomial Kernels for Planar F-deletionFedor V. Fomin, Daniel Lokshtanov and Saket Saurabh
  3. 3. Polynomial Kernels for Planar F-deletion * Fedor V. Fomin, Daniel Lokshtanov and Saket Saurabhwhen F contains connected graphs
  4. 4. Outline
  5. 5. OutlineéêçäçÖìÉ
  6. 6. OutlineëâÉíÅÜÉë=çÑ=âÉó=áÇÉ~ë
  7. 7. OutlineÉéáäçÖìÉ
  8. 8. Polynomial  Kernels  for  Planar  F-­deletion
  9. 9. Polynomial  Kernels  for  Planar  F-­deletion qÜÉ=píçêó=pç=c~ê
  10. 10. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå
  11. 11. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå |G|  p(k)?
  12. 12. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åçInfer the existence of a protrusion
  13. 13. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åçInfer the existence of a protrusion no  protrusions? Reject the instance
  14. 14. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åçInfer the existence of a protrusion no  protrusions? Reduce the protrusion Reject the instance
  15. 15. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åçInfer the existence of a protrusion no  protrusions? Reduce the protrusion Reject the instance
  16. 16. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åç óÉëInfer the existence of a protrusion Poly Kernel no  protrusions? Reduce the protrusion Reject the instance
  17. 17. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åç óÉëInfer the existence of a protrusion Poly Kernel Lemmas  17-­23 no  protrusions? Reduce the protrusion Reject the instance
  18. 18. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åç óÉëInfer the existence of a protrusion Poly Kernel Lemmas  17-­23 no  protrusions? Reduce the protrusion Reject the instance Theorem  2,3
  19. 19. mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åç óÉëInfer the existence of a protrusion Poly Kernel Lemmas  17-­23 no  protrusions? Reduce the protrusion Reject the instance Theorem  2,3
  20. 20. mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åç óÉëInfer the existence of a protrusion Poly Kernel no  protrusions? Reduce the protrusion Reject the instance Theorem  2,3
  21. 21. mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åç óÉëInfer the existence of a protrusion Poly Kernel no  protrusions? Reduce the protrusion Reject the instance
  22. 22. mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åç óÉëInfer the existence of a protrusion Infer  near-­protrusions Poly Kernel no  protrusions? Reduce the protrusion Reject the instance
  23. 23. mä~å~ê=cJÇÉäÉíáçå |G|  p(k)? åç óÉëInfer the existence of a protrusion Infer  near-­protrusions Poly Kernel no  protrusions? Irrelevant  Edges Reduce the protrusion Reject the instance
  24. 24. XGX
  25. 25. Approximate  F-­deletion  set XGX
  26. 26. Approximate  F-­deletion  set XGX constant  treewidth  zone
  27. 27. Approximate  F-­deletion  set XGX constant  treewidth  zone
  28. 28. Approximate  F-­deletion  set XGX constant  treewidth  zone
  29. 29. Approximate  F-­deletion  set X constant  boundary...?GX constant  treewidth  zone
  30. 30. kk
  31. 31. kk
  32. 32. The  Space  of  all  t-­boundaried  graphs
  33. 33. The  Space  of  all  t-­boundaried  graphs
  34. 34. The  Space  of  all  t-­boundaried  graphs
  35. 35. Polynomial  Kernels  for  Planar  F-­deletion
  36. 36. Polynomial  Kernels  for  Planar  F-­deletion `çåíáåìÉÇKKK
  37. 37. Approximate  F-­deletion  set X constant  boundary...?GX constant  treewidth  zone
  38. 38. Approximate  F-­deletion  set X constant  boundary...?GX constant  treewidth  zone
  39. 39. Approximate  F-­deletion  set X constant  boundary...?GX constant  treewidth  zone
  40. 40. Approximate  F-­deletion  set X constant  boundary...?GX constant  treewidth  zone
  41. 41. Approximate  F-­deletion  set X constant  boundary...?GX constant  treewidth  zone
  42. 42. Approximate  F-­deletion  set X constant  boundary...?GX constant  treewidth  zone
  43. 43. For  every  guess  we  have  a  protrusion
  44. 44. But  it  may  not  be  safe  to  reduce  them!
  45. 45. ëíê~íÉÖó=çìíäáåÉ
  46. 46. ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ
  47. 47. Suppose  the  protrusion  gives  us  a  way  of finding  irrelevant  edges. ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ
  48. 48. If  every  guess  declares  an  edge  to  be  irrelevant, then  it  is  safe  to  remove  it  from  G. ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ
  49. 49. If  every  guess  declares  an  edge  to  be  irrelevant, then  it  is  safe  to  remove  it  from  G. ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉIf  there  are  more  than  poly(k)  edges  incident  on  a  single   vertex,  then  one  of  them  is  not  relevant  to  any  guess.
  50. 50. Deleting  an  edge  never  increases  OPT.
  51. 51. Deleting  an  edge  never  increases  OPT. G
  52. 52. Deleting  an  edge  never  increases  OPT. G G{e}
  53. 53. Deleting  an  edge  never  increases  OPT. G G{e}
  54. 54. Deleting  an  edge  never  increases  OPT. G G{e} Determine  why  G  is  a  no-­instance, and  don’t  interfere  with  it.
  55. 55. G  does  not  have  a Vertex  Cover of  size 2
  56. 56. G  does  not  have  a Vertex  Cover of  size 2
  57. 57. G  does  not  have  a Vertex  Cover of  size 2
  58. 58. G  does  not  have  a Vertex  Cover of  size 2
  59. 59. G  does  not  have  a Vertex  Cover of  size 2
  60. 60. G  does  not  have  a Vertex  Cover of  size 2
  61. 61. G  does  not  have  a F-­deletion  Set of  size 2
  62. 62. G  does  not  have  a F-­deletion  Set of  size k
  63. 63. G  does  not  have  a F-­deletion  Set of  size k G  does  not  belong  to  [FDel]k
  64. 64. G  does  not  have  a F-­deletion  Set of  size k G  does  not  belong  to  [FDel]k But  [FDel]k  is  closed  under   minors,  and  hence  has  a  finite   obstruction  set  S.
  65. 65. G  does  not  have  a F-­deletion  Set of  size k G  does  not  belong  to  [FDel]k But  [FDel]k  is  closed  under   minors,  and  hence  has  a  finite   obstruction  set  S.S  “witnesses”  the  fact  that  G  is  a  NO  instance.  Edges  not  involved  in  copies  of  S  are...  irrelevant!
  66. 66. We  don’t  know  the  obstruction  sets.
  67. 67. We  don’t  know  the  obstruction  sets.Even  if  we  did,  minor  models  of  graphs  in   S  could  be  arbitrarily  large.
  68. 68. Approximate  F-­deletion  set XGX constant  treewidth  zone
  69. 69. Approximate  F-­deletion  set XGX constant  treewidth  zone
  70. 70. Approximate  F-­deletion  set X GX constant  treewidth  zone
  71. 71. Approximate  F-­deletion  set X GX constant  treewidth  zone
  72. 72. Approximate  F-­deletion  set X  GX constant  treewidth  zone
  73. 73. Before Approximate  F-­deletion  set X  GX constant  treewidth  zone
  74. 74. Before After Approximate  F-­deletion  set X  GX constant  treewidth  zone
  75. 75. Before After Approximate  F-­deletion  set XGX constant  treewidth  zone
  76. 76. Before After Approximate  F-­deletion  set XGX constant  treewidth  zone
  77. 77. Before After Approximate  F-­deletion  set XGX constant  treewidth  zone
  78. 78. ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ
  79. 79. Avoiding  some  obstruction  to  membership  in  FDelk ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ
  80. 80. Avoiding  some  obstruction  to  membership  in  FDelkLarge  degree  implies  the  existence  of  at  least  one  irrelevant  edge ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ
  81. 81. Avoiding  some  obstruction  to  membership  in  FDelk Large  degree  implies  the  existence  of  at  least  one  irrelevant  edgeUse  near-­protrusions,  cost  vectors,  finite  index,  CMSO  expressibility. ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ
  82. 82. X Approximate  F-­deletion  setGX constant  treewidth  zone:  tw=c
  83. 83. X Approximate  F-­deletion  setGX constant  treewidth  zone:  tw=c
  84. 84. X Approximate  F-­deletion  set Case  1GX There  is  a  (k+c+1)-­sized (u,v)-­separator. constant  treewidth  zone:  tw=c
  85. 85. X Approximate  F-­deletion  set Case  1GX There  is  a  (k+c+1)-­sized (u,v)-­separator. constant  treewidth  zone:  tw=c
  86. 86. X Approximate  F-­deletion  set Case  1 GX There  is  a  (k+c+1)-­sized (u,v)-­separator.bounded  by  poly(k) constant  treewidth  zone:  tw=c
  87. 87. X Approximate  F-­deletion  set Case  2 GX There  is  a  (k+c+1)  flow between  u  and  v.bounded  by  poly(k) constant  treewidth  zone:  tw=c
  88. 88. X Approximate  F-­deletion  set Case  2 GX There  is  a  (k+c+1)  flow between  u  and  v.bounded  by  poly(k) constant  treewidth  zone:  tw=c
  89. 89. GS Case  2 There  is  a  (k+c+1)  flow between  u  and  v.bounded  by  poly(k) constant  treewidth  zone:  tw=c
  90. 90. GS Case  2 There  is  a  (k+c+1)  flow between  u  and  v.bounded  by  poly(k) constant  treewidth  zone:  tw=c
  91. 91. GS Case  2 There  is  a  (k+c+1)  flow between  u  and  v.bounded  by  poly(k) constant  treewidth  zone:  tw=c
  92. 92. GS Case  2 There  is  a  (k+c+1)  flow between  u  and  v.bounded  by  poly(k) constant  treewidth  zone:  tw=c
  93. 93. a  separator  of  size  (c+1)+k GS Case  2 There  is  a  (k+c+1)  flow between  u  and  v.bounded  by  poly(k) constant  treewidth  zone:  tw=c
  94. 94. Polynomial  Kernels  for Planar F-­deletion
  95. 95. Polynomial  Kernels  for Planar F-­deletion qÜÉ=`ÉåëçêÉÇ=aÉí~áäë
  96. 96. The  presence  of  disconnected  graphs  in  F  opens  up  a  can  of  worms,  various  details   need  substantial  tweaking.
  97. 97. The  finite  obstructions  are  implied  by  WQO  of  t-­boundaried  graphs  of  bounded  treewidth  with  special  minor  operations.
  98. 98. Consequences  of  the  kernel.Obstructions  to  [FDel]k  are  polynomially   bounded  in  k.
  99. 99. Polynomial  Kernels  for Planar F-­deletion
  100. 100. Polynomial  Kernels  for F-­deletion
  101. 101. Polynomial  Kernels  for F-­deletion qç=_É=`çåíáåìÉÇKKK
  102. 102. Daniel  will  answer  your  questions!
  103. 103. Daniel  will  answer  your  questions!

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