1. Bart Jansen Polynomial Kernels for Hard Problems on Disk Graphs
SWAT 2010, Bergen

2. 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

3. 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 (unless …)
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, …

4. 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 (unless …)
Have polynomial kernels in (unit)disk graphs
And the problems are still hard on disk graphs

5. 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)

6. 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

7. 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

8. Connected vertex cover

9. 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 (E.J. van Leeuwen [Thesis])
We study the problem on disk graphs

10. 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, and live otherwise
Dead vertices do not cover additional edges, but connect vertices
Marked
Live unmarked
Dead unmarked
Unmarked

11. Reduction Rules
For unmarked vertex v, let NC(v) be the connected components of G[S] that v is adjacent to
Unmarked vertex v with degree > k: mark v
Distinct dead vertices u,v with NC(u) ⊆ NC(v): delete u
Afterwards: sets NC(v) for dead vertices v form an antichain X Y Z W
Marked
Live unmarked
Dead unmarked

12. Analysis To prove: |V| is O(k2) in reduced instances
Count vertices in three sets
marked vertices in set S
live, unmarked vertices
dead, unmarked vertices

13. Analysis To prove: |V| is O(k2) in reduced instances
Count vertices in three sets
marked vertices in set S
live, unmarked vertices
dead, unmarked vertices
If |S| > k then we output NO
there is no connected vertex cover of size ≤k containing S

14. Analysis To prove: |V| is O(k2) in reduced instances
Count vertices in three sets
marked vertices in set S ≤k
live, unmarked vertices
dead, unmarked vertices

15. Analysis To prove: |V| is O(k2) in reduced instances
Count vertices in three sets
marked vertices in set S ≤k
live, unmarked vertices
dead, unmarked vertices
Call an edge between live, unmarked vertices an uncovered edge
A live unmarked vertex is incident on
≥ 1 uncovered edges (definition of live)
≤ k uncovered edges (Rule 1)
If there are more than k2 uncovered edges, output NO
Otherwise # live vertices is ≤ 2k2

16. Analysis To prove: |V| is O(k2) in reduced instances
Count vertices in three sets
marked vertices in set S ≤k
live, unmarked vertices ≤2k2
dead, unmarked vertices

17. Analysis To prove: |V| is O(k2) in reduced instances
Count vertices in three sets
marked vertices in set S ≤k
live, unmarked vertices ≤2k2
dead, unmarked vertices
Use the geometry (relationship to planar graph)
We consider the disks of the dead vertices and marked vertices

18. Bounding dead vertices
Dead vertices are non-adjacent (by definition)
Consider regions induced by connected components of marked vertices
Marked
Dead unmarked

19. Bounding dead vertices
Subtract interior of dead disks from the connected regions
Label the regions, use them as vertices in a planar graph
Add completely overlapped regions
Vertex for each dead disk, adjacent to touching regions V C B A W X Y Z
Marked
Dead unmarked

20. Bounding dead vertices
Bipartite planar graph on dead vertices D and regions R
At most 1 region in R per marked vertex
Neighborhoods of vertices in D form an antichain
Planarity implies: |D| ≤ 5|R| ≤ 5|S| ≤ 5k V C B A W X Y Z

21. Analysis
To prove: |V| is O(k2) in reduced instances
Count vertices in three sets
marked vertices in set S ≤ k
live, unmarked vertices ≤ 2k2
dead, unmarked vertices ≤ 5k
Summing up: |V| ≤ 2k2 + 6k
We can undo the annotation at the cost of k(k+1) extra vertices
Connected Vertex Cover on disk graphs has a kernel with 3k2 + 7k vertices

22. other results

23. H-matching
Input: Graph G, connected graph H, integer k
Question: Does G contain k vertex-disjoint subgraphs isomorphic to H?
Parameter: k
NP-complete when H is a triangle, even on planar graphs
In FPT on general graphs for fixed H
Kernel with O(k|H|-1) vertices by H. Moser [SOFSEM ‘09]
No kernel polynomial in |H| + k (unless …)
We consider the case where G is a unit-disk graph, H is arbitrary
H-matching on unit-disk graphs has a kernel with O(k) vertices for fixed H

24. 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|)

25. 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|)
Parameterization by k is W[1] hard on unit-disk graphs
Parameterization by min(|R|,|B|) is FPT on general graphs
No polynomial kernel unless… (Dom et al. [ICALP 2009])
Red/Blue Dominating Set has a
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

26. Conclusion and discussion
Several parameterized problems without polynomial kernels on general graphs, do allow polynomial kernels on dense (unit)disk graphs
Polynomial kernels for Red/Blue Dom. Set and Connected V.C. in Ki,j-subgraph-free graphs
Open problems
Linear-vertex kernel for Connected VC in disk graphs?
Polynomial kernel for H-matching in disk graphs?
Polynomial kernel for unit-disk Edge Clique Cover?
Polynomial kernel for unit-disk Partition (Vertex Set) Into Cliques?
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