From FVS to F-Deletion

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From FVS to F-Deletion

  1. 1. From FVS to F-deletion
  2. 2. From FVS to F-deletion a simple constant-factor randomized approximation algorithm
  3. 3. From VC to F-deletion a simple constant-factor randomized approximation algorithm
  4. 4. A Generic Algorithm
  5. 5. A Generic Algorithm Special Cases
  6. 6. THE BLUEPRINT
  7. 7. EveryVertex Cover intersects every edgeat at least one endpoint.
  8. 8. Every Solution intersects every edgeat at least one endpoint.
  9. 9. Every Solution intersectssome subset of edges? at at least one endpoint.
  10. 10. Every Solution intersectsa good fraction of edges at at least one endpoint.
  11. 11. Every Solution intersectsa good fraction of edges at at least one endpoint.
  12. 12. Pick an edge e, uniformly at random.
  13. 13. Pick an endpoint of e, uniformly at random.
  14. 14. Repeat until a solution is obtained.
  15. 15. Pick an edge e, uniformly at random.Pick an endpoint of e, uniformly at random.Repeat until a solution is obtained.
  16. 16. Pick an edge e, uniformly at random. Pick a good edge with probability (1/c)Pick an endpoint of e, uniformly at random.Repeat until a solution is obtained.
  17. 17. Pick an edge e, uniformly at random. Pick a good edge with probability (1/c)Pick an endpoint of e, uniformly at random. Pick a good endpoint with probability (1/2)Repeat until a solution is obtained.
  18. 18. Pick an edge e, uniformly at random. Pick a good edge with probability (1/c)Pick an endpoint of e, uniformly at random. Pick a good endpoint with probability (1/2)Repeat until a solution is obtained. The expected solution size: 2c(OPT)
  19. 19. Pick an edge e, uniformly at random. Pick a good edge with probability (1/c)Pick an endpoint of e, uniformly at random. Pick a good endpoint with probability (1/2)Repeat until a solution is obtained. The expected solution size: 2c(OPT)
  20. 20. SGS
  21. 21. SGS
  22. 22. SGS #cross edges + #edges within S (1/c) · m
  23. 23. SGS #cross edges + #edges within S (1/c) · m  P v2S d(v) 2
  24. 24. SGS P v2S d(v) (1/c) · m 2
  25. 25. SGS X d(v) (1/c) · m ·2 v2S
  26. 26. SGS X d(v) (1/c) · m ·2 v2S
  27. 27. SGS X d(v) (1/c) · m ·2 v2S X X d(v) (1/c) · d(v) v2S v2G
  28. 28. SGS X X d(v) (1/c) · d(v) v2S v2G
  29. 29. SPECIAL CASES
  30. 30. GS isan independent set.
  31. 31. GS isa matching
  32. 32. SGS Preprocess: Delete isolated edges. X X d(v) (1/c) · d(v) v2S v2G
  33. 33. SGS Preprocess: Delete isolated edges. X X d(v) (1/4) · d(v) v2S v2G
  34. 34. GS isan acyclic graph (forest)
  35. 35. GS isan acyclic graph (forest)cÉÉÇÄ~Åâ=sÉêíÉñ=pÉí
  36. 36. SGS Preprocess: ??? X X d(v) (1/c) · d(v) v2S v2G
  37. 37. SWhen can we say that every leaf “contributes” a cross-edge?
  38. 38. SPreprocess: Delete pendant vertices.
  39. 39. #of cross edges #of leaves
  40. 40. #of cross edges #of leaves#of edges in the tree = #of leaves + #internal nodes - 1
  41. 41. #of leaves #internal nodes (minimum degree at least three) #of cross edges #of leaves#of edges in the tree = #of leaves + #internal nodes - 1
  42. 42. #of leaves #internal nodes (minimum degree at least three) #of cross edges #of leaves#of edges in the tree #of leaves + #of leaves -1
  43. 43. #of leaves #internal nodes (minimum degree at least three) #of cross edges #of leaves#of edges in the tree #of leaves + #of leaves -1#of edges in the tree 2(#of leaves) -1
  44. 44. #of leaves #internal nodes (minimum degree at least three) #of cross edges #of leaves#of edges in the tree #of leaves + #of leaves -1#of edges in the tree 2(#of leaves) -1#of edges in the tree 2(#of cross edges) -1
  45. 45. #of edges in the tree 2(#of cross edges) -1
  46. 46. #of edges in the tree 2(#of cross edges) -1 X X d(v) (1/c) · d(v) v2S v2G
  47. 47. #of edges in the tree 2(#of cross edges) -1 X X d(v) (1/c) · d(v) v2S v2GX d(v) = 2(#of edges in the tree) + 2(#of cross edges)v2G
  48. 48. #of edges in the tree 2(#of cross edges) -1 X X d(v) (1/c) · d(v) v2S v2GX d(v) = 2(#of edges in the tree) + 2(#of cross edges)v2G 2(2#cross edges - 1) + 2(#of cross edges)
  49. 49. #of edges in the tree 2(#of cross edges) -1 X X d(v) (1/c) · d(v) v2S v2GX d(v) = 2(#of edges in the tree) + 2(#of cross edges)v2G 2(2#cross edges - 1) + 2(#of cross edges) 6(#cross edges)
  50. 50. #of edges in the tree 2(#of cross edges) -1 X X d(v) (1/c) · d(v) v2S v2GX d(v) = 2(#of edges in the tree) + 2(#of cross edges)v2G 2(2#cross edges - 1) + 2(#of cross edges) 6(#cross edges) ! X 6 d(v) v2S
  51. 51. #of leaves #internal nodes(minimum degree at least three)
  52. 52. #of leaves #internal nodes(minimum degree at least three)
  53. 53. #of leaves #internal nodes(minimum degree at least three) More preprocessing!
  54. 54. #of leaves #internal nodes (minimum degree at least three) #of cross edges 2(#of leaves)#of edges in the tree = #of leaves + #of leaves -1#of edges in the tree 2(#of leaves) -1#of edges in the tree 2(#of cross edges) -1
  55. 55. #of leaves #internal nodes (minimum degree at least three) #of cross edges 2(#of leaves)#of edges in the tree = #of leaves + #of leaves -1#of edges in the tree 2(#of leaves) -1#of edges in the tree 2(#of cross edges) -1#of edges in the tree #of cross edges -1
  56. 56. GS is independentGS is a matchingGS is acyclic
  57. 57. GS is independent Factor 2, for free.GS is a matchingGS is acyclic
  58. 58. GS is independent Factor 2, for free. Factor 4, after removingGS is a matching isolated edgesGS is acyclic
  59. 59. GS is independent Factor 2, for free. Factor 4, after removingGS is a matching isolated edges Factor 4, after deleting degree 1GS is acyclic and short-circuiting degree 2 vertices.
  60. 60. WHAT’S NEXT?
  61. 61. What is the most general problem for which the algorithm “just works”?
  62. 62. Beyond problem-specific reduction rules...Is there a one-size-fits-all?
  63. 63. Answer: mä~å~ê=cJÇÉäÉíáçå
  64. 64. Remove at most k vertices such that theremaining graph has no minor models of graphs from F.
  65. 65. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã Remove at most k vertices such that the remaining graph has no minor models of graphs from F.
  66. 66. 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.)
  67. 67. Independent = no edges Forbid an edge as a minor
  68. 68. Acyclic = no cycles Forbid a triangle as a minor
  69. 69. Pathwidth-one graphs Forbid T2, K3 as a minor
  70. 70. Turns out that when you want to kill minor models of planar graphs,GS must have bounded treewidth.
  71. 71. This can be exploited to framesome very general reduction rules.
  72. 72. This can be exploited to framesome very general reduction rules. http://arxiv.org/abs/1204.4230
  73. 73. A brief summary of this discussion http://neeldhara.com/planar-f-deletion-1/ Thank You!
  74. 74. A brief summary of this discussion http://neeldhara.com/planar-f-deletion-1/ Thank You!

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