Kernels for Planar F-Deletion (Restricted Variants)

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Kernels for Planar F-Deletion (Restricted Variants)

  1. 1. KERNELSFOR F-DELETION
  2. 2. aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ=cJÇÉäÉíáçå=éêçÄäÉã
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  4. 4. ^ééêçñáã~íáçå= dÉííáåÖ=ÅäçëÉ=íç ~å=çéíáã~ä=cJÜáííáåÖ=ëÉí aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ=cJÇÉäÉíáçå=éêçÄäÉã líÜÉê=RÉëìäíë= ^å=lîÉêîáÉï
  5. 5. ^ééêçñáã~íáçå= `çåÅäìÇáåÖ dÉííáåÖ=ÅäçëÉ=íç RÉã~êâë= ~å=çéíáã~ä=cJÜáííáåÖ=ëÉí aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ=cJÇÉäÉíáçå=éêçÄäÉã líÜÉê=RÉëìäíë= ^å=lîÉêîáÉï
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  8. 8. KERNELIZATION
  9. 9. A kernelization procedure ⇤ ⇤is a function f : {0, 1} N ⇥ {0, 1} N such that for all (x, k), |x| = n (f (x, k)) 2 L i (x, k) 2 L 0 0 |x | = g(k) and k k and f is polynomial time computable.
  10. 10. The F-Deletion Problem
  11. 11. A classic optimization questionoften takes the following general form...
  12. 12. A classic optimization question often takes the following general form...How “close” is a graph to having a certain property?
  13. 13. This question can be formalized in a number of ways, and a well-studied version is the following:
  14. 14. What is the smallest number of vertices that need to be deleted so that the remaining graph is __________________?
  15. 15. What is the smallest number of vertices that need to be deleted so that the remaining graph is independent?
  16. 16. What is the smallest number of vertices that need to be deleted so that the remaining graph is acyclic?
  17. 17. What is the smallest number of vertices that need to be deleted so that the remaining graph is planar?
  18. 18. What is the smallest number of vertices that need to be deleted so that the remaining graph is constant treewidth?
  19. 19. What is the smallest number of vertices that need to be deleted so that the remaining graph is in X?
  20. 20. X = a property
  21. 21. A property = an infinite collection of graphs
  22. 22. that satisfy the property.A property = an infinite collection of graphs
  23. 23. that satisfy the property.A property = an infinite collection of graphs can often be characterized by a finite set of forbidden minors
  24. 24. that satisfy the property. A property = an infinite collection of graphswhenever the family is closed under minors, Graph Minor Theorem can often be characterized by a finite set of forbidden minors
  25. 25. Independent = no edges Forbid an edge as a minor
  26. 26. Acyclic = no cycles Forbid a triangle as a minor
  27. 27. Planar Graphs Forbid a K3,3, K5 as a minor
  28. 28. Pathwidth-one graphs Forbid T2, K3 as a minor
  29. 29. Remove at most k vertices such that theremaining graph has no minor models of graphs from F.
  30. 30. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F.
  31. 31. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. NP-Complete (Lewis, Yannakakis)
  32. 32. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. NP-Complete FPT (Lewis, Yannakakis) (Robertson, Seymour)
  33. 33. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. Polynomial Kernels NP-Complete FPT (Lewis, Yannakakis) (Robertson, Seymour)
  34. 34. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. Polynomial Kernels? NP-Complete FPT (Lewis, Yannakakis) (Robertson, Seymour)
  35. 35. mä~å~êqÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. Polynomial Kernels? NP-Complete FPT (Lewis, Yannakakis) (Robertson, Seymour)
  36. 36. mä~å~êqÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. (Where F contains a planar graph.) Polynomial Kernels? NP-Complete FPT (Lewis, Yannakakis) (Robertson, Seymour)
  37. 37. mä~å~êqÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. (Where F contains a planar graph.) Polynomial Kernels? NP-Complete FPT (Lewis, Yannakakis) (Robertson, Seymour) Remark. We assume throughout that F contains connected graphs.
  38. 38. A Summary of Results
  39. 39. A Summary of Results• Planar F-deletion admits an approximation algorithm.
  40. 40. A Summary of Results• Planar F-deletion admits an approximation algorithm.• Planar F-deletion admits a polynomial kernel on claw-free graphs.
  41. 41. A Summary of Results• Planar F-deletion admits an approximation algorithm.• Planar F-deletion admits a polynomial kernel on claw-free graphs.• Planar F-deletion admits a polynomial kernel whenever F contains the “onion” graph.
  42. 42. A Summary of Results• Planar F-deletion admits an approximation algorithm.• Planar F-deletion admits a polynomial kernel on claw-free graphs.• Planar F-deletion admits a polynomial kernel whenever F contains the “onion” graph.• The “disjoint” version of the problem admits a kernel.
  43. 43. A Summary of Results• Planar F-deletion admits an approximation algorithm.• Planar F-deletion admits a polynomial kernel on claw-free graphs.• Planar F-deletion admits a polynomial kernel whenever F contains the “onion” graph.• The “disjoint” version of the problem admits a kernel.• The onion graph admits an Erdős–Pósa property.
  44. 44. A Summary of Results• Planar F-deletion admits an approximation algorithm.• Planar F-deletion admits a polynomial kernel on claw-free graphs.• Planar F-deletion admits a polynomial kernel whenever F contains the “onion” graph.• The “disjoint” version of the problem admits a kernel.• The onion graph admits an Erdős–Pósa property.• Some packing variants of the problem are not likely to have polynomial kernels.
  45. 45. A Summary of Results• Planar F-deletion admits an approximation algorithm.• Planar F-deletion admits a polynomial kernel on claw-free graphs.• Planar F-deletion admits a polynomial kernel whenever F contains the “onion” graph.• The “disjoint” version of the problem admits a kernel.• The onion graph admits an Erdős–Pósa property.• Some packing variants of the problem are not likely to have polynomial kernels.• The kernelization complexity of Independent FVS and Colorful Motifs is explored in detail.
  46. 46. ^ééêçñáã~íáçå= `çåÅäìÇáåÖ dÉííáåÖ=ÅäçëÉ=íç RÉã~êâë= ~å=çéíáã~ä=cJÜáííáåÖ=ëÉí aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ=cJÇÉäÉíáçå=éêçÄäÉã líÜÉê=RÉëìäíë= ^å=lîÉêîáÉï
  47. 47. ^ééêçñáã~íáçå= `çåÅäìÇáåÖ dÉííáåÖ=ÅäçëÉ=íç RÉã~êâë= ~å=çéíáã~ä=cJÜáííáåÖ=ëÉí aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ=cJÇÉäÉíáçå=éêçÄäÉã líÜÉê=RÉëìäíë= ^å=lîÉêîáÉï
  48. 48. qÜÉ=mä~å~ê=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. The graphs in F are connected, and at least one of them is planar.
  49. 49. Ingredients
  50. 50. 1. Let H be a planar graph on h vertices. If the treewidth of G exceeds ch then G contains a minor model of H.2. The planar F-deletion problem can be solved optimally in polynomial time on graphs of constant treewidth. 3. Any YES instance of planar F-deletion has treewidth at most k + ch .
  51. 51. Constant treewidth Large enough to guarantee a minor model of H, but still aconstant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2)
  52. 52. Constant treewidth Large enough to guarantee a minor model of H, but still aconstant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
  53. 53. “Small” Separator Bounded in terms of k (Fact 3)Constant treewidth Large enough to guarantee a minor model of H, but still aconstant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
  54. 54. “Small” Separator Bounded in terms of k (Fact 3)Constant treewidth Large enough to guarantee a minor model of H, but still aconstant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
  55. 55. “Small” Separator Bounded in terms of k (Fact 3)Constant treewidth Large enough to guarantee a minor model of H, but still aconstant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
  56. 56. “Small” Separator Bounded in terms of k (Fact 3) Solve Optimally Large enough to guarantee a minor model of H, but still aconstant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
  57. 57. “Small” Separator Bounded in terms of k (Fact 3) Solve Optimally Large enough to guarantee a minor model of H, but still aconstant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) Recurse
  58. 58. ~å~äóëáë
  59. 59. ~å~äóëáë
  60. 60. ~å~äóëáë
  61. 61. ~å~äóëáë
  62. 62. ~å~äóëáë
  63. 63. ~å~äóëáë
  64. 64. ~å~äóëáë
  65. 65. at most k ~å~äóëáë
  66. 66. ~å~äóëáë at most k at most k at most kat most k at most k at most k at most k
  67. 67. ~å~äóëáë at most k at most k at most kat most k poly(n) at most k poly(n) at most k poly(n) at most k poly(n) poly(n)
  68. 68. How do we get here?
  69. 69. 1. Let H be a planar graph on h vertices. If the treewidth of G exceeds ch then G contains a minor model of H.2. The planar F-deletion problem can be solved optimally in polynomial time on graphs of constant treewidth. 3. Any YES instance of planar F-deletion has treewidth at most k + ch .
  70. 70. 1. Let H be a planar graph on h vertices. If the treewidth of G exceeds ch then G contains a minor model of H.2. The planar F-deletion problem can be solved optimally in polynomial time on graphs of constant treewidth. 3. Any YES instance of planar F-deletion has treewidth at most k + ch .
  71. 71. pk log k
  72. 72. pk log k
  73. 73. Repeat.
  74. 74. pThe solution size is proportional to k 2 log k
  75. 75. p The solution size is proportional to k 2 log kCan be improved to k(log k)3/2 with the help of bootstrapping.
  76. 76. Running the algorithm throughvalues of k between 1 and n (starting from 1) leads to an approximationfor the optimization version of the problem.
  77. 77. ^ééêçñáã~íáçå= `çåÅäìÇáåÖ dÉííáåÖ=ÅäçëÉ=íç RÉã~êâë= ~å=çéíáã~ä=cJÜáííáåÖ=ëÉí aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ=cJÇÉäÉíáçå=éêçÄäÉã líÜÉê=RÉëìäíë= ^å=lîÉêîáÉï
  78. 78. ^ééêçñáã~íáçå= `çåÅäìÇáåÖ dÉííáåÖ=ÅäçëÉ=íç RÉã~êâë= ~å=çéíáã~ä=cJÜáííáåÖ=ëÉí aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ=cJÇÉäÉíáçå=éêçÄäÉã líÜÉê=RÉëìäíë= ^å=lîÉêîáÉï
  79. 79. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. Polynomial Kernels? NP-Complete FPT (Lewis, Yannakakis) (Robertson, Seymour)
  80. 80. Conjecture
  81. 81. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F.
  82. 82. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F.The problem admits polynomial kernels when F contains a planar graph.
  83. 83. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. On Claw free graphsThe problem admits polynomial kernels when F contains a planar graph.
  84. 84. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. particularThe problem admits polynomial kernels when F contains a planar graph.
  85. 85. Protrusion-based reductions the idea
  86. 86. A Boundary of Constant SizeConstant Treewidth
  87. 87. A Boundary of Constant SizeConstant Treewidth
  88. 88. A Boundary of Constant SizeConstant Treewidth
  89. 89. The space of t-boundaried graphscan be broken up into equivalence classes based on how they “behave” with the “other side” of the boundary.
  90. 90. The value of theoptimal solution is the sameup to a constant.
  91. 91. The space of t-boundaried graphscan be broken up into equivalence classes based on how they “behave” with the “other side” of the boundary.
  92. 92. The space of t-boundaried graphs can be broken up into equivalence classes based on how they “behave” with the “other side” of the boundary. For some problems, the number of equivalence classes is finite,allowing us to replace protrusions in graphs.
  93. 93. For the protrusion-based reductions to take effect, we require subgraphs of constant treewidth that are separated from the rest of the graph by a constant-sized separator.
  94. 94. Approximation AlgorithmFor the protrusion-based reductions to take effect, we require subgraphs of constant treewidth that are separated from the rest of the graph by a constant-sized separator.
  95. 95. F-hitting Set Constant Treewidth
  96. 96. Approximation AlgorithmFor the protrusion-based reductions to take effect, we require subgraphs of constant treewidth that are separated from the rest of the graph by a constant-sized separator.
  97. 97. Approximation Algorithm For the protrusion-based reductions to take effect, we require subgraphs of constant treewidth that are separated from the rest of the graph by a constant-sized separator.Restrictions like claw-freeness.
  98. 98. ^ééêçñáã~íáçå= `çåÅäìÇáåÖ dÉííáåÖ=ÅäçëÉ=íç RÉã~êâë= ~å=çéíáã~ä=cJÜáííáåÖ=ëÉí aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ=cJÇÉäÉíáçå=éêçÄäÉã líÜÉê=RÉëìäíë= ^å=lîÉêîáÉï
  99. 99. ^ééêçñáã~íáçå= `çåÅäìÇáåÖ dÉííáåÖ=ÅäçëÉ=íç RÉã~êâë= ~å=çéíáã~ä=cJÜáííáåÖ=ëÉí aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ=cJÇÉäÉíáçå=éêçÄäÉã líÜÉê=RÉëìäíë= ^å=lîÉêîáÉï
  100. 100. crRqebR=afRb`qflkp
  101. 101. crRqebR=afRb`qflkp• What happens when we drop the planarity assumption?
  102. 102. crRqebR=afRb`qflkp• What happens when we drop the planarity assumption?• What happens if there are graphs in the forbidden set that are not connected?
  103. 103. crRqebR=afRb`qflkp• What happens when we drop the planarity assumption?• What happens if there are graphs in the forbidden set that are not connected?• Are there other infinite classes of graphs (not captured by finite sets of forbidden minors) for which the same reasoning holds?
  104. 104. crRqebR=afRb`qflkp• What happens when we drop the planarity assumption?• What happens if there are graphs in the forbidden set that are not connected?• Are there other infinite classes of graphs (not captured by finite sets of forbidden minors) for which the same reasoning holds?• How do structural requirements on the solution (independence, connectivity) affect the complexity of the problem?
  105. 105. ^`hkltibadjbkqp
  106. 106. ^`hkltibadjbkqp Abhimanyu M. Ambalath, S. Arumugam, Radheshyam Balasundaram, K. Raja Chandrasekar, Michael R. Fellows, Fedor V. Fomin,Venkata Koppula, Daniel Lokshtanov, Matthias Mnich N. S. Narayanaswamy, Geevarghese Philip,Venkatesh Raman, M. S. Ramanujan, Chintan Rao H., Frances A. Rosamond, Saket Saurabh, Somnath Sikdar, Bal Sri Shankar
  107. 107. Thank you!

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