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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?