1. Bart Jansen Polynomial Kernels for Hard Problems on Disk Graphs
Accepted for presentation at SWAT 2010
TACO Day, June 10th 2010, Aachen

2. Overview Introduction Kernels Conclusion Kernelization Graph classes
Triangle Packing, Kt-matching, H-matching
Red/Blue Dominating Set
Connected Vertex Cover
Conclusion

3. Kernelization for graph problems
Consider a computational decision problem on graphs
Input: encoding x of a question about graph G and integer k.
Question: does graph G have a (…)?
Parameter:k
Parameter expresses some property of the question (size of what we are looking for, treewidth of graph, etc.)
A kernelization algorithm takes (x, k) as input and computes instance (x’, k’) of same problem in polynomial time, such that
Answer to x is YES  answer to x’ is YES
k’ ≤ g(k) for some function g
|x’| ≤ f(k) for some function f
The function f is the size of the kernel
We want f to be a (small) polynomial

4. Recent kernelization results
Bad news
Good news
Many parameterized problems are W[1]-hard and have no kernels
Several easier parameterized problems only have kernels where f is exponential
If we require G to be planar, lots of problems have linear or quadratic kernels
Even if we relax planarity to bounded genus, H-minor-free, …

5. Expanding the range of good news
The frameworks giving general good news about small kernels only apply under restrictions that make the graph G sparse: |E| ≤ c |V|
Dense graphs without special structure make the problem hard, implying non-existence of kernels
We consider graphs that exhibit structure, but are not sparse: (unit)disk graphs
Yields good news:
Red-Blue Dominating Set, H-Matching, Connected Vertex Cover
Do not have polynomial kernels in general graphs
Have polynomial kernels in (unit)disk graphs
And the problems are still hard on disk graphs

6. Subquadratic edge count Kernels for Dominating Set
Graph classes
Linear edge count
Quadratic edge count
Meta-theorems
Our kernels
planar
unit-disk
bounded-genus
bounded-genus
H-minor-free
disk
Ki,j-subgraph-free
Subquadratic edge count
Kernels for Dominating Set
general

7. Disk graphs Consider a set S of closed disks in the plane
The intersection graph of S:
has a vertex v for every disk D(v),
has an edge between u and v iff. the disks D(v) and D(u) intersect.
(touching disks also intersect)

8. Properties of disk graphs
If all disks have the same radius, their intersection graph is a unit disk graph
All planar graphs are disk graphs (varying radii)
Any clique is a (unit)disk graph
Compare with K5 which is not planar
So there are disk graphs with
Class of (unit)disk graphs
Closed under vertex deletion
Not closed under edge deletion
Not closed under edge contraction

9. Triangle packing and h-matching
Structure theory and kernels
Triangle packing and h-matching

10. Triangle Packing Input: Graph G, integer k
Question: Are there k vertex-disjoint triangles in G?
Parameter: k
NP-complete, even on planar graphs
In FPT on general graphs with a O(k2)-vertex kernel

11. Triangle Packing Input: Graph G, integer k
Question: Are there k vertex-disjoint triangles in G?
Parameter: k
Single reduction rule
Try all O(n3) sets of size 3, and test if they form a triangle
Mark vertices that occur in a triangle
Delete all vertices that were not marked

12. Kernelization algorithm
Greedily build a maximal triangle packing
Suppose the greedy packing contains k* copies
If k* ≥ k
The problem is solved
Output a trivial YES instance
If k* < k
We prove:
|V| is O(k*) is O(k)

13. Neighborhood Clique Lemma
Let v be a vertex in a unit-disk graph G. Then there is a clique of size ⌈deg(v) / 6⌉ among the neighbors of G.
G[N(v)] has a clique of size ⌈deg(v) / 6⌉
Proof.
Consider centers of v and its neighbors in a disk realization
Divide the plane into 6 equal sectors around v
Some sector contains ⌈deg(v) / 6⌉ vertices (Pigeonhole Principle) v

14. Neighbors in each sector form a clique
Assume every disk has radius ½
If v has a neighbor x then distance |xv| ≤ 1 y v x v

15. Neighbors in each sector form a clique
Assume every disk has radius ½
If v has a neighbor x then distance |xv| ≤ 1
Consider two neighbors x,y in the same sector
By adjacency to v: |xv| ≤ 1, |yv| ≤ 1
Sector definition: angle xvy ≤ 60o
By law of Cosines: |xy| ≤ 1
So x,y adjacent
Neighbors within sector form a clique y x v

16. Analysis of kernel size
If there is a maximal triangle packing with k* copies in G, then |V| is O(k*)
Proof.
We divide V in two subsets:
set S with vertices that are used in a selected copy
set W with the remainder
Since all triangles are vertex-disjoint, there are exactly 3k* vertices in S (every triangle uses 3 vertices)
We bound the size of W
Every vertex in W must be adjacent to vertex in S
Every vertex in S has at most 12 neighbors in W
So |W| ≤ 12 |S| ≤ 12(3 k*) ∈ O(k*)

17. Extension to Kt-matching
We get a kernel with O(k) vertices for Triangle Packing in unit-disk graphs
Current best kernel for general graphs has O(k2) vertices
Generalizes to Kt-matching for every fixed t
Pack vertex-disjoint complete subgraphs of size t
Important properties still hold:
Every vertex that is not selected in a maximal packing must be adjacent to a selected vertex
Every selected vertex has O(t) neighbors in W

18. Extension to H-Matching
H-matching problem
Pack vertex-disjoint copies of a fixed connected graph H
Kernel with O(k|H|-1) vertices by H. Moser [SOFSEM ‘09]
No kernel polynomial in |H| + k
H-matching on unit-disk graphs
H can be arbitrary
Graph G in which we find the copies is a unit-disk graph
Our result
O(k)-vertex kernel for every fixed graph H
Constant is exponential in the diameter of H
Properties of maximal H-matching in reduced graph
Every unused vertex has distance ≤ diameter(H) to a used vertex
Every vertex has O(|H|) unused neighbors

19. Red/blue dominating set
Structure theory and kernels
Red/blue dominating set

20. Red/Blue Dominating Set
Input: Graph G with red vertices R, blue vertices B, integer k
Question: Is there a set of ≤ k red vertices that dominate all blue vertices?
Parameter: min(|R|,|B|)

21. Background
min(|R|,|B|) as parameter since parameter k is W[1] hard, even on unit-disk graphs
In FPT on general graphs, no polynomial kernel
Usually assume G is bipartite with R and B as color classes
We do not assume this here; bipartite disk graphs are planar
Our results:
O(min(|R|,|B|))-vertex kernel on planar graphs
O(min(|R|,|B|)2)-vertex kernel on unit-disk graphs
O(min(|R|,|B|)4)-vertex kernel on disk graphs

22. Reduction Rules Red vertices r1, r2 such that N(r1) ∩ B ⊆ N(r2) ∩ B
Delete r1
Blue vertices b1, b2 such that N(b1) ∩ R ⊆ N(b2) ∩ R
Delete b2

23. Balance
After exhaustive application of reduction rules, the color classes must be balanced
Number of vertices in the classes must be polynomially related
Easy for planar graphs: |R| ≤ 5|B| (and vice versa)
Contribution:
|R| ∈ O(|B|2) (and vice versa) for unit-disk graphs
|R| ∈ O(|B|4) (and vice versa) for disk graphs
These structural results immediately yield kernels

24. Balance in colored unit-disk graphs
Usual model: two vertices adjacent iff their disks intersect
Double the radius of disks
Now: two vertices adjacent iff the disk of one contains the center of the other, and vice versa
We prove: if no two red vertices see the same blue vertices, then |R| ∈ O(|B|2). ∙
radius 1 ∙
radius ½

25. Proof
We prove: if no two red vertices see the same blue vertices, then |R| ∈ O(|B|2)
Look at arrangement of the plane induced by blue circles
Each region contains at most one red center
Complexity of the arrangement is O(|B|2) ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙

26. Balance in colored disk graphs
Reconsider usual model: vertices adjacent iff disks intersect
We prove: if no red disk sees a subset of the blue vertices seen by another red disk, then |R| ∈ O(|B|4)
[A,B]
[A,B,C]
[B,A,C]
[B,A] A B C

27. Balance in colored disk graphs
A face in the arrangement of bisector curves determines a unique order of encountering blue disks
The blue neighbors of a red disk are a prefix of the string determined by the face containing its center
So any face contains at most one red disk
[A,B,C]
[B,A,C] A B C
[B,C,A]
[A,C,B]
[C,A,B]
[C,B,A]

28. Balance in colored disk graphs
Given n curves for which each pair intersects O(1) times, the complexity of the arrangement is O(n2)
We have O(|B|2) curves, hence complexity is O(|B|4)
Total number of red disks is O(|B|4)
[A,B,C]
[B,A,C]
[B,C,A]
[A,C,B]
[C,A,B]
[C,B,A]

29. Summary of kernels for Red/Blue Dominating Set
By applying the reduction rules we find in polynomial time an equivalent instance such that no red vertex sees a subset of what another red vertex sees
Same for the blue vertices
Structural theorems show that in such colored graphs the sizes of the color classes are polynomially related
So size of the largest class is polynomial in the size of smallest class
Hence |V| = |R| + |B| ≤ min(|R|+|B|) + max(|R|,|B|) is O(min(|R|+|B|)c)

30. Connected vertex cover
Structure theory and kernels
Connected vertex cover

31. Connected Vertex Cover
Input: Graph G, integer k
Question: Is there a vertex cover of ≤ k vertices that induces a connected subgraph?
Parameter: k
FPT on general graphs, no polynomial kernel
Trivial linear-vertex kernel on unit-disk graphs
Any vertex cover for a connected unit-disk graph must have size ≥ n/12 (Erik Jan van Leeuwen’s thesis)

32. Annotated Connected Vertex Cover
Input: Graph G, set of marked vertices S, integer k
Question: Is there a vertex cover of ≤ k vertices that induces a connected subgraph, and which contains all marked vertices?
Parameter: k
Unmarked vertex v is dead if all its neighbors are marked, if not then v is live
Reduction rules
Unmarked vertex v with degree > k: mark v
Distinct dead vertices u,v such that N(u) ⊆ N(v): delete u

33. Analysis Call an edge covered if it’s incident on a marked vertices
Otherwise an edge is uncovered
> k2 uncovered edges: output NO
> k marked vertices: output NO
In remaining cases ≤ k2 uncovered edges
≤ 2k2 live vertices since each live vertex is incident on an uncovered edge
≤ k marked vertices
Remains to bound the dead vertices
# dead vertices can be bounded in # marked vertices by the balance argument, gives #dead is O(k4)
More intricate argument gives O(k2) bound
Annotation can be undone

34. Conclusion and discussion
Several parameterized problems without polynomial kernels on general graphs, do allow polynomial kernels on dense (unit)disk graphs
Colored Ki,j-subgraph-free graphs also have the “polynomial balance property”
Polynomial kernels for Red/Blue Dom. Set and Connected V.C.
Open problems
Poly kernel for H-matching in disk graphs?
Poly kernel for unit-disk Edge Clique Cover?
Poly kernel for unit-disk Partition (Vertex Set) Into Cliques?
Improve the quartic bound for balance in disk graphs
Find other problems where colored graph balance implies poly kernels
Planar
Unit-disk
Disk
H-matching
O(k) [Known]
O(k) ?
Red/Blue Dominating Set
O(min(|R|,|B|))
O(min(|R|,|B|)2)
O(min(|R|,|B|)4)
Connected Vertex Cover
14k [Known]
12k
3k2 + 7k