1. Fixed parameter tractability II: Kernelization
Algorithms & Networks
Today: Thomas van Dijk

2. Overview Fixed parameter complexity recap Two kernelizations
Vertex Cover again
Cluster Editing
Aspects of kernelization: Feedback Vertex Set
Algorithm
Analysis
Implementation
Experimentation

3. Quick reintroduction to FPT / Kernelization

4. Hard problems ‘Natural’ problems tend to be either
in P  Nice
NP-Hard  Nasty
How to solve nasty problems?
Solve only small instances
Don’t actually solve the problem: … approximate … Exploit properties of certain instances

5. Fixed parameter complexity
In many applications, some number can be assumed to be small
Time of algorithm can be exponential in this small number, but should be polynomial in usual size of problem

6. Fixed parameter problems
Given: Graph G, integer k, …
Parameter: k
Question: Does G have a ??? of size at least (at most) k?
Examples: vertex cover, independent set, coloring, …
Examples
k-Coloring NP-complete for k>2…
k-Clique Seems to require W( nf(k) ) time…
k-Vertex Cover Solvable in O( 2k (n+m) ) time.

7. Fixed parameter complexity theory
To distinguish between behavior:
O( f(k) poly(n) )  Fixed Parameter Tractable
O( n f(k) )  Not so nice
Proposed by Downey and Fellows.

8. Kernelization algorithms
Kernelization = “Preprocessing with quality guarantee”
Polynomial time algorithm:
Input: Graph G, parameter k
Output: Graph G’, parameter k’ < k,
Such that
( G’, k’ ) is YES iff ( G, k ) is YES
G’ has O( f(k) ) vertices

9. Kernel iff FPT Kernelization  FPT FPT  Kernelization Kernelize.
Solve kernel by any exact algorithm.
Runtime: O( pkernelize-time(n) + fsolve-time(fkernel-size(k)) )
FPT  Kernelization
Definition of FPT: has O( p(n) f(k) ) time algorithm
If n < f(k), input is kernel.
If n > f(k), solve problem with FPT algorithm; output some YES or NO instance accordingly.

10. Typical form of kernelization
Repeat some rules, until no rule is possible
Rules can do some necessary modification and decrease k.
Rules can remove some part of the graph.
Rules can output YES or NO.
Sometimes add ‘annotation’ to the graph

11. Kernel for Vertex Cover

12. Vertex cover: simple kernel
Rule 1: if there is a vertex v of deg(v)>k, then remove v, decrease k by one.
Rule 1 is safe.
Consider v not in vertex cover.
Covering v’s edges costs >k vertices.
Resulting instance is NO.
Therefore, if instance is YES, then v is in the cover.
Therefore, instance is YES iff converted instance is YES.

13. Vertex cover: simple kernel
Rule 2: if not rule 1, and m > k2, then answer NO.
Rule 2 is safe.
By [not rule 1], no vertex has degree > k.
Therefore, a vertex covers at most k edges.
Therefore, instance with > k2 edges must be NO.
Analysis of kernel size
By rule 2, m ≤ k2.
In general, n ≤ 2m.
Analysis of runtime: clearly polynomial

14. Kernel for Cluster Editing

15. Cluster editing Instance: undirected graph G=(V,E), integer k
Parameter: k
Question: can we make at most k modifications to G, such that each connected component is a clique, by
adding edges
deleting edges
O( k2 ) vertices
Gramm, Guo, Hüffner, Niedermeier
CIAC’03
O( k ) vertices
E.g. Fellows, Langston, Rosamund, Shaw
FCT’07

16. Trivial rules and a plan
Rule 1: If a connected component of G is a clique, remove this connected component
Rule 2: If we have more than k connected components and Rule 1 does not apply: answer NO
Consequence: after Rule 1 and Rule 2, there are at most k connected components
Plan: find rules that make connected component small
Annotate the graph: pairs of vertices can be permanent or forbidden.

17. Observation and rule 3
If two vertices have k+1 neighbors in common, they must belong to the same clique in a solution
Rule 3: Suppose vertices v, w have k+1 neighbors in common. If the edge did not exist, add it and decrease k by 1 Set the edge {v,w} to be permanent

18. Another observation and rule 4
If there are at least k+1 vertices that are adjacent to exactly one of v and w, then {v,w} cannot be an edge in the solution
Rule 4: Suppose vertex v has k+1 neighbors that do not neighbor vertex w If {v,w} is an edge: delete it and decrease k by one Mark the pair {v,w} as forbidden
Rule 5: if a pair is both forbidden and permanent then there is no solution

19. Transitivity (triangles)
Rule 6: if {v,w} is permanent, and {w,x} is permanent, then set {w,x} to be permanent
if the edge did not exist, add it, and decrease k by one
Rule 7: if {v,w} is permanent and {w,x} is forbidden, then set {w,x} to be forbidden
if the edge existed, delete it, and decrease k by one

20. Runtime Rules can be executed in polynomial time
With properly chosen data structures, can kernelize a graph in O(n3) time

21. Analysis: plan of attack
Already know: at most k connected components
Small components are fine
So suppose component is big … we’ll show contradiction
We’ll aim at a quadratic kernel

22. Analysis: counting
Consider a connected component C with at least 4k+1 vertices.
Let K = vertices not involved in a modification
K, by definition, must already form a clique
Since k modifications cannot touch > 2k vertices: |K| ≥ 2k+1
All edges in K already are permanent by rule 3

23. Counting continued Consider a vertex v in C-K
v is adjacent to k+1 vertices in K
By rule 3, v gets permanent edges to all of K
That is, v joins K.
v is non-adjacent to k+1 vertices in K
By rule 4, v gets forbidden edges to all of K
That is, v leaves all of K.
Since |K|>2k+1, any v fits one of the above cases.
Therefore, each connected component has size at most 4k
In total: at most 4k2 vertices

24. Fixed parameter complexity of feedback set problems

25. Overview Feedback set problems Cubic kernel for Feedback Vertex Set
Algorithmic considerations
Experimental evaluation

26. Feedback Vertex Set Instance: Question: An undirected graph G = (V,E)
An integer k
Question:
Does there exist an S V of size at most k such that G[ V-S ] is acyclic?

27. Example

28. Example: an FVS

29. Example: an FVS

30. Example: not an FVS

31. Example: not an FVS

32. Example: a lowerbound

33. Directed feedback vertex set (and Loop Cutset)
Feedback set problems
Feedback vertex set
Feedback edge set
Feedback arc set
Directed feedback vertex set (and Loop Cutset)

34. Feedback edge set Easy: spanning tree
Feedback set problems
Feedback vertex set
Feedback edge set Easy: spanning tree
Feedback arc set
Directed feedback vertex set (and Loop Cutset)

35. NP-COMPLETE Feedback set problems
Feedback vertex set
Feedback edge set Easy: spanning tree
NP-COMPLETE
Feedback arc set
Directed feedback vertex set (and Loop Cutset)

36. Feedback Vertex Set Instance: Parameter: Question:
An undirected graph G = (V,E)
Parameter:
An integer k
Question:
Does there exist an S V of size at most k such that G[ V-S ] is acyclic?

37. Directed feedback vertex set (and Loop Cutset)
Feedback set problems
Feedback vertex set
Feedback edge set Easy
NP-COMPLETE
Feedback arc set
Directed feedback vertex set (and Loop Cutset)

38. Feedback set problems Parameterized on size of solution
Feedback vertex set
Fixed parameter tractable
Feedback edge set Easy
Feedback arc set
Directed feedback vertex set
(and Loop Cutset)

39. Feedback set problems Parameterized on size of solution
Feedback vertex set
(and Loop Cutset)
Fixed parameter tractable
Feedback edge set Easy
Feedback arc set
Directed feedback vertex set

40. Feedback set problems Parameterized on size of solution
Feedback vertex set
Directed feedback vertex set
(and Loop Cutset)
Fixed parameter tractable
Feedback edge set Easy
and Loop Cutset ?
Feedback arc set

41. Intuition: “why” FVS is FPT
For n→∞, fixed k, instances become either
Easy
“Because” they are sparse
Complicated but in a way that is easy to recognize
“There is no way this can be done with only k vertices!”
It is hard to make any complicated structure without making lots of cycles

42. Feedback Vertex Set is FPT
Almost directly from some deep theorems of fixed parameter complexity
Not practical
There are several FPT algorithms, e.g.
O*( 5k )
Chen, Fomin, Liu, Lu, Villanger
O*( 4k ) probabilistic
Becker, Bar-Yehuda, Geiger
Practical…?

43. Kernels for feedback vertex set
O(k11) vertices
Burrage, Estivill-Castro, Fellows, Langston, Mac & Rosamond
IWPEC’06
O(k3) vertices
Bodlaender
STACS’06
O(k2) vertices
Thomassé
SODA’09

44. Kernelization overview
A collection of rules
A proof: if none of the rules apply, then the graph can have only O( k3 ) vertices
Complicated; not in this presentation
An algorithm:
just keep trying the rules until none apply

45. Simple rules Islet rule: Twig rule: Triple edge rule:
Remove degree 0 vertices.
Twig rule:
Remove degree 1 vertices.
Triple edge rule:
Remove more-than-double edges.

46. Degree two rule Degree 2 rule: Suppose v has degree 2. Bypass v.
There exists optimal FVS that does not use v: choose one of its neighbors

47. Self-loops
Self-loop rule: Suppose v has a self-loop. Remove v and decrease k by one.
Vertex v must be in the FVS.

48. Large double degree
Large double degree rule: Suppose v has k+1 double edges. Remove v and decrease k by one.

49. Flower rule example, k ≤ 2

50. Flower rule example, k ≤ 2 v

51. Flower rule example, k ≤ 2

52. Flower rule
Flower rule: Suppose there are k+1 cycles that are vertex-disjoint except all include v. Remove v and decrease k by one.
If v is not chosen in the FVS then all k+1 cycles need to be broken separately.

53. Cascade of degree-2 rule

54. Cascade of degree-2 rule

55. Cascade of degree-2 rule

56. Cascade of degree-2 rule

57. Cascade of degree-2 rule

58. Improvement rule example, k ≤ 2B

59. Improvement rule example, k ≤ 2B

60. Improvement rule
Improvement rule: Suppose there are k+2 vertex-disjoint paths between vertices u and v. Add a double edge (u,v).
Any two of the paths form a cycle.
If neither u nor v is chosen, at least k+1 of the paths must be broken separately.
So at least one of u and v must be chosen.

61. Improvement rule example, k ≤ 2B

62. Improvement rule example, k ≤ 1B

63. Improvement rule example, k ≤ 1B

64. Large double degree, k ≤ 1

65. Large double degree, k ≤ 1

66. 2-Approximation rule
2-Approximation rule: Suppose a 2-approximate FVS is larger than 2k. Conclude NO.
This can be found in polynomial time
Becker, Geiger
Bafna, Berman, Fujito

67. Rule: Abdication Suppose a vertex v governs a piece X.
If it has one edge into X, remove that edge.
If it has multiple edges into X, select v in the FVS.

68. Rule overview Islet, Twig, Triple edge Degree two Self-loop, Flower
Improvement
2-Approximation
Abdication

69. Analysis Let A be a 2-approximate FVS.
By 2-Approximation rule: |A| ≤ 2k
Let B be vertices with a double edge to A.
By Large Double Degree rule: |B| in O(k2)
Call a connected components of G[V-A-B] a piece.
We will bound the number of pieces by O(k3).
More analysis gives O(k3) vertices and edges. Skipped here.

70. Analysis Border of a piece: its neighbors in A and B.
Associate each piece with a pair of vertices in its border that does not have a double edge
Exists for each piece, by Twig and Abdication.
Case distinction on the type of those two vertices
Case both in A:
More than k+2 such pieces would give Improvement.
O( k2 ) pairs of vertices in A, at most k+1 pieces each.
O( k3 ) pieces of this type.

71. Analysis
Case both in B
No two such pieces: would be an unbroken cycle.
Ω( k4 ) pairs of vertices in B; not done yet.
We know A is feedback vertex set:
B is not in the feedback vertex set.
Pieces are not in the feedback vertex set.
Vertices in B and how they are connected by pieces: must be a forest!
O( k2 ) vertices in B; therefore O( k2 ) pieces of this type.

72. Analysis Case one in A, one in B O( k3 ) such pairs
If only one piece for the pair: done.
Consider the pairs (a,b) that do have multiple pieces associated to it.
For any a, at most k such b: Flower rule.
For any (a,b), at most k+1 pieces: Improvement rule.
k vertices in A, each paired with at most k vertices in B, each with at most k+1 pieces: O( k3 ) such pieces.
QED

73. Related results Quadratic kernel Linear kernel on planar FVS
Thomassé
SODA’09
Linear kernel on planar FVS
Bodlaender & Penninkx
IWPEC’08
Directed FVS is FPT
Chen, Liu, Lu & O’Sullivan, Razgon
STOC’08

74. Algorithmic considerations
A bunch of rules … are they all “required?”
The complicated ones are.
Dropping any one of those: size of output no longer bounded in k.
The O(k3) size bound, is it tight?
Yes: exists reduced instance with Θ(k3) vertices
How to efficiently check for the rules?

75. Order of rule checking Matters a lot in an implementation
Some aspects:
Cascading small-degree rules.
Improvement makes double edges, leads to Large Double Degree
Abdication rule uses pieces. Only available when 2-approximation is up-to-date.

76. Justification of the rules
The Flower rule is required

77. Justification of the rules
The Improvement rule is required
The Abdication rule is required

78. Justification of the rules
The 2-Approximation rule is required
Consider a grid
Degree-2 rule cuts the corners

79. Justification of the rules
The 2-Approximation rule is required
Consider a grid
Nothing else happens if k ≥ 3

80. Justification of the rules
The 2-Approximation rule is required
Consider a grid
Nothing else happens if k ≥ 3:
not Flower (just 2 petals)

81. Justification of the rules
The 2-Approximation rule is required
Consider a grid
Nothing else happens if k ≥ 3:
not Improvement (just 4 paths)

82. Justification of the rules
The 2-Approximation rule is required
Consider a grid
Only the 2-approximation

83. Checking for the improvement rule?

84. Checking for the improvement rule
Node-capacity flow
Standard reduction to arc-capacity flow

85. Checking for the Flower rule?

86. Checking for the Flower rule
If v is a flower with k petals, there is a selection of edges where:
v has 2k of its edges selected
All other vertices have either 2 or none of their edges selected

87. Flower with 3 petals 2 2 2 2 2 2 6 2 2 2 2 2 2

88. Checking for the Flower rule
Reduce to perfect matching in general graphs
Then “just” use Edmonds’ algorithm

89. Flower with 3 petals 2 2 2 2 2 2 6 2 2 2 2 2 2

90. {0,2}-Matchting widget
No edges selected (vertex not in petal)

91. {0,2}-Matchting widget
Two edges selected (vertex in petal)

92. Implementation C++ : Interesting experimental results
Library of Efficient Data structures and Algorithms
By ‘Algorithmic Solutions’
Quite nice
Interesting experimental results

93. Runtime on random graphs, k=20

94. Runtime on random graphs, k=20

95. Runtime on random graphs

96. Effectiveness of the kernelization
After kernelization, one still needs to solve the problem on the kernel
So compare runtime of
solve, versus
kernelize and solve kernel
Following data: solve using a simple branch-and-bound

97. Effectiveness of the kernelization

98. Effective branching factor (histogram)
With kernelization
Without kernelization

99. UAI’06 Networks: runtime

100. UAI’06 Networks: effectiveness as heuristic (histogram)

101. Effectiveness on UAI’06 Networks
Unfortunately, those are way too big to find an optimal loop cutset for
Thousands of nodes …
Don’t actually need optimal loop cutset for cutset conditioning
Kernelization improves quality of heuristics?

102. Random graphs

103. UAI’06 Networks

104. Some results from the experiments
The kernelization runs quickly
Kernelization helps practical exact runtime a lot!
Kernelization helps the Becker/Geiger 2-approximation quality a lot!
As a heuristic, seems much more practical than Becker/Bar-Yehuda/Geiger O(4k) algorithm

105. Kernelization for Loop Cutset

106. Loop Cutset Instance: Parameter: Question:
An directed acyclic graph G = (V,A)
Parameter:
An integer k
Question:
Does there exist an S V of size at most k such that for all simple loops, S contains a vertex that is not a ‘sink’ on that loop

107. Loop Cutset Instance: Parameter: Question:
An directed acyclic graph G = (V,A)
Parameter:
An integer k
Question:
Does there exist an S V of size at most k such that for all simple loops, S contains a vertex that is not a ‘sink’ on that loop

108. Kernelization for Loop Cutset
Transform to Blackout FVS
Kernelize Blackout FVS
Based on Bodlaender’s FVS kernelization
Transform back

109. Blackout Feedback Vertex Set
Instance:
An undirected graph G=(V,E)
A set X V of ‘blacked out’ vertices
Parameter:
An integer k
Question:
Does there exist an S V-X of size at most k such that G[ V-S ] is acyclic

110. Transform to Blackout FVS
Split all vertices v into vin and vout
Blackout all in vertices

111. Transform to Blackout FVS
Split all vertices v into vin and vout
Blackout all in vertices

112. Transform to Blackout FVS
Split all vertices v into vin and vout
Blackout all in vertices

113. Some additional rules
Degree two rule becomes somewhat more complicated
Handle the blacked out vertices…

114. Transform back to Loop Cutset
Possibly edges between
Two allowed vertices
An allowed and a blacked out vertex
Not between two blacked out vertices
Blacked out component rule