1. Polynomial Kernels for
Planar F-deletion

2. Polynomial Kernels for
Planar F-deletion
Fedor V. Fomin, Daniel Lokshtanov and Saket Saurabh

3. Polynomial Kernels for
Planar F-deletion *
Fedor V. Fomin, Daniel Lokshtanov and Saket Saurabh
when F contains connected graphs

4. Outline

5. Outline
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6. Outline
ëâÉíÅÜÉë=çÑ=âÉó=áÇÉ~ë

7. Outline
ÉéáäçÖìÉ

8. Polynomial Kernels for Planar F-­deletion

9. Polynomial Kernels for Planar F-­deletion
qÜÉ=píçêó=pç=c~ê

10. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå

11. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?

12. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç
Infer the existence of a protrusion

13. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç
Infer the existence of a protrusion
no protrusions?
Reject the instance

14. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç
Infer the existence of a protrusion
no protrusions?
Reduce the protrusion Reject the instance

15. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç
Infer the existence of a protrusion
no protrusions?
Reduce the protrusion Reject the instance

16. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç óÉë
Infer the existence of a protrusion Poly Kernel
no protrusions?
Reduce the protrusion Reject the instance

17. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç óÉë
Infer the existence of a protrusion Poly Kernel
Lemmas 17-­23
no protrusions?
Reduce the protrusion Reject the instance

18. RÉëíêáÅíÉÇ=`~ëÉë=çÑ=mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç óÉë
Infer the existence of a protrusion Poly Kernel
Lemmas 17-­23
no protrusions?
Reduce the protrusion Reject the instance
Theorem 2,3

19. mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç óÉë
Infer the existence of a protrusion Poly Kernel
Lemmas 17-­23
no protrusions?
Reduce the protrusion Reject the instance
Theorem 2,3

20. mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç óÉë
Infer the existence of a protrusion Poly Kernel
no protrusions?
Reduce the protrusion Reject the instance
Theorem 2,3

21. mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç óÉë
Infer the existence of a protrusion Poly Kernel
no protrusions?
Reduce the protrusion Reject the instance

22. mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç óÉë
Infer the existence of a protrusion
Infer near-­protrusions Poly Kernel
no protrusions?
Reduce the protrusion Reject the instance

23. mä~å~ê=cJÇÉäÉíáçå
|G|  p(k)?
åç óÉë
Infer the existence of a protrusion
Infer near-­protrusions Poly Kernel
no protrusions?
Irrelevant Edges
Reduce the protrusion Reject the instance

24. X
GX

25. Approximate F-­deletion set
X
GX

26. Approximate F-­deletion set
X
GX
constant treewidth zone

27. Approximate F-­deletion set
X
GX
constant treewidth zone

28. Approximate F-­deletion set
X
GX
constant treewidth zone

29. Approximate F-­deletion set
X
constant boundary...?
GX
constant treewidth zone

36. k
k

37. k
k

40. The Space of all t-­boundaried graphs

41. The Space of all t-­boundaried graphs

42. The Space of all t-­boundaried graphs

43. Polynomial Kernels for Planar F-­deletion

44. Polynomial Kernels for Planar F-­deletion
`çåíáåìÉÇKKK

45. Approximate F-­deletion set
X
constant boundary...?
GX
constant treewidth zone

46. Approximate F-­deletion set
X
constant boundary...?
GX
constant treewidth zone

47. Approximate F-­deletion set
X
constant boundary...?
GX
constant treewidth zone

48. Approximate F-­deletion set
X
constant boundary...?
GX
constant treewidth zone

49. Approximate F-­deletion set
X
constant boundary...?
GX
constant treewidth zone

50. Approximate F-­deletion set
X
constant boundary...?
GX
constant treewidth zone

51. For every guess we have a protrusion

52. But it may not be safe to reduce them!

54. ëíê~íÉÖó=çìíäáåÉ

55. ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ

56. Suppose the protrusion gives us a way of
finding irrelevant edges.
ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ

57. If every guess declares an edge to be irrelevant,
then it is safe to remove it from G.
ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ

58. If every guess declares an edge to be irrelevant,
then it is safe to remove it from G.
ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ
If there are more than poly(k) edges incident on a single
vertex, then one of them is not relevant to any guess.

60. Deleting an edge never increases OPT.

61. Deleting an edge never increases OPT.
G

62. Deleting an edge never increases OPT.
G G{e}

63. Deleting an edge never increases OPT.
G G{e}

64. Deleting an edge never increases OPT.
G G{e}
Determine why G is a no-­instance,
and don’t interfere with it.

65. G does not have a Vertex Cover of size 2

66. G does not have a Vertex Cover of size 2

67. G does not have a Vertex Cover of size 2

68. G does not have a Vertex Cover of size 2

69. G does not have a Vertex Cover of size 2

70. G does not have a Vertex Cover of size 2

71. G does not have a F-­deletion Set of size 2

72. G does not have a F-­deletion Set of size k

73. G does not have a F-­deletion Set of size k
G does not belong to [FDel]k

74. 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.

75. 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!

77. We don’t know the obstruction sets.

78. We don’t know the obstruction sets.
Even if we did, minor models of graphs in
S could be arbitrarily large.

84. Approximate F-­deletion set
X
GX
constant treewidth zone

85. Approximate F-­deletion set
X
GX
constant treewidth zone

86. Approximate F-­deletion set
X

GX
constant treewidth zone

87. Approximate F-­deletion set
X

GX
constant treewidth zone

88. Approximate F-­deletion set
X
 
GX
constant treewidth zone

89. Before
Approximate F-­deletion set
X
 
GX
constant treewidth zone

90. Before After
Approximate F-­deletion set
X
 
GX
constant treewidth zone

91. Before After
Approximate F-­deletion set
X
GX
constant treewidth zone

92. Before After
Approximate F-­deletion set
X
GX
constant treewidth zone

93. Before After
Approximate F-­deletion set
X
GX
constant treewidth zone

94. ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ

95. Avoiding some obstruction to membership in FDelk
ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ

96. Avoiding some obstruction to membership in FDelk
Large degree implies the existence of at least one irrelevant edge
ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ

97. Avoiding some obstruction to membership in FDelk
Large degree implies the existence of at least one irrelevant edge
Use near-­protrusions, cost vectors, finite index, CMSO expressibility.
ë~ÑÉíó=ö=ëáòÉ=ö=íáãÉ

98. X
Approximate F-­deletion set
GX
constant treewidth zone: tw=c

99. X
Approximate F-­deletion set
GX
constant treewidth zone: tw=c

100. X
Approximate F-­deletion set
Case 1
GX There is a (k+c+1)-­sized
(u,v)-­separator.
constant treewidth zone: tw=c

101. X
Approximate F-­deletion set
Case 1
GX There is a (k+c+1)-­sized
(u,v)-­separator.
constant treewidth zone: tw=c

102. 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

103. 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

104. 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

105. GS
Case 2
There is a (k+c+1) flow
between u and v.
bounded by poly(k)
constant treewidth zone: tw=c

106. GS
Case 2
There is a (k+c+1) flow
between u and v.
bounded by poly(k)
constant treewidth zone: tw=c

107. GS
Case 2
There is a (k+c+1) flow
between u and v.
bounded by poly(k)
constant treewidth zone: tw=c

108. GS
Case 2
There is a (k+c+1) flow
between u and v.
bounded by poly(k)
constant treewidth zone: tw=c

109. 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

110. Polynomial Kernels for Planar F-­deletion

111. Polynomial Kernels for Planar F-­deletion
qÜÉ=`ÉåëçêÉÇ=aÉí~áäë

113. The presence of disconnected graphs in
F opens up a can of worms, various details
need substantial tweaking.

114. The finite obstructions are implied by
WQO of t-­boundaried graphs of bounded
treewidth with special minor operations.

115. Consequences of the kernel.
Obstructions to [FDel]k are polynomially
bounded in k.

116. Polynomial Kernels for Planar F-­deletion

117. Polynomial Kernels for F-­deletion

118. Polynomial Kernels for F-­deletion
qç=_É=`çåíáåìÉÇKKK

119. Daniel will answer your questions!

120. Daniel will answer your questions!