Kernels for Planar F-Deletion (Restricted Variants)
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The document discusses kernelization for the F-deletion problem, where graphs in F are connected and at least one is planar. It is shown that the planar F-deletion problem admits a polynomial kernel whenever F contains a planar graph called the "onion" graph. Several other positive and negative results are also presented, including that planar F-deletion admits an approximation algorithm and a polynomial kernel on claw-free graphs. The document concludes by outlining the ingredients for showing that planar F-deletion admits a polynomial kernel.
Kernels for Planar F-Deletion (Restricted Variants)
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- 12. 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.
- 17. A classic optimization question often takes the following general form...
- 18. A classic optimization question often takes the following general form... How “close” is a graph to having a certain property?
- 19. This question can be formalized in a number of ways, and a well-studied version is the following:
- 20. What is the smallest number of vertices that need to be deleted so that the remaining graph is __________________?
- 21. What is the smallest number of vertices that need to be deleted so that the remaining graph is independent?
- 22. What is the smallest number of vertices that need to be deleted so that the remaining graph is acyclic?
- 23. What is the smallest number of vertices that need to be deleted so that the remaining graph is planar?
- 24. What is the smallest number of vertices that need to be deleted so that the remaining graph is constant treewidth?
- 25. What is the smallest number of vertices that need to be deleted so that the remaining graph is in X?
- 26. X = a property
- 27. A property = an infinite collection of graphs
- 28. that satisfy the property. A property = an infinite collection of graphs
- 29. that satisfy the property. A property = an infinite collection of graphs can often be characterized by a finite set of forbidden minors
- 30. that satisfy the property. A property = an infinite collection of graphs whenever the family is closed under minors, Graph Minor Theorem can often be characterized by a finite set of forbidden minors
- 31. Independent = no edges Forbid an edge as a minor
- 32. Acyclic = no cycles Forbid a triangle as a minor
- 33. Planar Graphs Forbid a K3,3, K5 as a minor
- 34. Pathwidth-one graphs Forbid T2, K3 as a minor
- 35. Remove at most k vertices such that the remaining graph has no minor models of graphs from F.
- 36. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F.
- 37. 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)
- 38. 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)
- 39. 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)
- 40. 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)
- 41. mä~å~ê 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)
- 42. mä~å~ê qÜÉ=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)
- 43. mä~å~ê qÜÉ=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.
- 45. A Summary of Results
- 46. A Summary of Results • Planar F-deletion admits an approximation algorithm.
- 47. A Summary of Results • Planar F-deletion admits an approximation algorithm. • Planar F-deletion admits a polynomial kernel on claw-free graphs.
- 48. 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.
- 49. 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.
- 50. 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.
- 51. 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.
- 52. 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.
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- 55. 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.
- 56. Ingredients
- 57. 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 .
- 59. Constant treewidth Large enough to guarantee a minor model of H, but still a constant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2)
- 60. Constant treewidth Large enough to guarantee a minor model of H, but still a constant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
- 61. “Small” Separator Bounded in terms of k (Fact 3) Constant treewidth Large enough to guarantee a minor model of H, but still a constant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
- 62. “Small” Separator Bounded in terms of k (Fact 3) Constant treewidth Large enough to guarantee a minor model of H, but still a constant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
- 63. “Small” Separator Bounded in terms of k (Fact 3) Constant treewidth Large enough to guarantee a minor model of H, but still a constant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
- 64. “Small” Separator Bounded in terms of k (Fact 3) Solve Optimally Large enough to guarantee a minor model of H, but still a constant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) The Rest of the Graph
- 65. “Small” Separator Bounded in terms of k (Fact 3) Solve Optimally Large enough to guarantee a minor model of H, but still a constant - so that the problem can be solved optimally in polynomial time. (Fact 1 & 2) Recurse
- 66. ~å~äóëáë
- 67. ~å~äóëáë
- 68. ~å~äóëáë
- 69. ~å~äóëáë
- 70. ~å~äóëáë
- 71. ~å~äóëáë
- 72. ~å~äóëáë
- 73. at most k ~å~äóëáë
- 74. ~å~äóëáë at most k at most k at most k at most k at most k at most k at most k
- 75. ~å~äóëáë at most k at most k at most k at most k poly(n) at most k poly(n) at most k poly(n) at most k poly(n) poly(n)
- 77. How do we get here?
- 78. 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 .
- 79. 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 .
- 81. p k log k
- 82. p k log k
- 84. Repeat.
- 85. p The solution size is proportional to k 2 log k
- 86. p The solution size is proportional to k 2 log k Can be improved to k(log k)3/2 with the help of bootstrapping.
- 87. Running the algorithm through values of k between 1 and n (starting from 1) leads to an approximation for the optimization version of the problem.
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- 90. 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)
- 91. Conjecture
- 92. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F.
- 93. 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.
- 94. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. On Claw free graphs The problem admits polynomial kernels when F contains a planar graph.
- 95. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F. particular The problem admits polynomial kernels when F contains a planar graph.
- 97. Protrusion-based reductions the idea
- 98. A Boundary of Constant Size Constant Treewidth
- 99. A Boundary of Constant Size Constant Treewidth
- 100. A Boundary of Constant Size Constant Treewidth
- 104. 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.
- 107. The value of the optimal solution is the same up to a constant.
- 108. 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.
- 109. 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.
- 110. 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.
- 111. 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.
- 112. F-hitting Set Constant Treewidth
- 113. 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.
- 114. 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.
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- 116. ^ééêçñáã~íáçå= `çåÅäìÇáåÖ dÉííáåÖ=ÅäçëÉ=íç RÉã~êâë= ~å=çéíáã~ä=cJÜáííáåÖ=ëÉí aÉÑáåáíáçåë= hÉêåÉäë=C=íÜÉ= cJÇÉäÉíáçå=éêçÄäÉã líÜÉê=RÉëìäíë= ^å=lîÉêîáÉï
- 117. crRqebR=afRb`qflkp
- 118. crRqebR=afRb`qflkp • What happens when we drop the planarity assumption?
- 119. 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?
- 120. 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?
- 121. 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?
- 122. ^`hkltibadjbkqp
- 123. ^`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
- 124. Thank you!