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# Bart Jansen Polynomial Kernels for Hard Problems on Disk Graphs - PowerPoint PPT Presentation

Bart Jansen Polynomial Kernels for Hard Problems on Disk Graphs. Accepted for presentation at SWAT 2010. Overview. Introduction Kernelization Graph classes Kernels Triangle Packing, K t -matching, H-matching Red/Blue Dominating Set Connected Vertex Cover Conclusion.

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### Bart JansenPolynomial Kernels for Hard Problems on Disk Graphs

Accepted for presentation at SWAT 2010

• Introduction

• Kernelization

• Graph classes

• Kernels

• Triangle Packing, Kt-matching, H-matching

• Red/Blue Dominating Set

• Connected Vertex Cover

• Conclusion

• 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

Good news

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, …

• Many parameterized problems are W[1]-hard and have no kernels

• Several easier parameterized problems only have kernels where f is exponential

• 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

Linear edge count

Meta-theorems

planar

unit-disk

bounded-genus

bounded-genus

H-minor-free

disk

Our kernels

Ki,j-subgraph-free

Kernels for Dominating Set

general

• 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)

• 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

Triangle packing and h-matching

• 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

• 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

• Greedily build a maximal triangle packing

• Suppose the greedy packing contains k* copies

• 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⌉ sectors (Pigeonhole Principle)

v

• Assume every disk has radius ½

• If v has a neighbor x then distance |xv| ≤ 1

y

v

x

v

• 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

• Neighbors within sector form a clique

y

x

v

• 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*)

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

• 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

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|)

• 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

• 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

• 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

• 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).

• 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)

• 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]

[B,A,C]

A

B

C

• 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

[B,C,A]

[A,C,B]

C

[C,A,B]

[C,B,A]

• 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]

• 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)

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 unit-disk graph must have size ≥ n/12 (Erik-Jan’s thesis)

• 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

• 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

• 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