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Bart Jansen

Polynomial Kernels for Hard Problems on Disk Graphs

Accepted for presentation at SWAT 2010

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Overview

Introduction Kernelization Graph classes

Kernels Triangle Packing, Kt-matching, H-matching Red/Blue Dominating Set Connected Vertex Cover

Conclusion

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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

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Recent kernelization results

Bad news Many parameterized

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

Several easier parameterized problems only have kernels where f is exponential

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, …

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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

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Our kernels

Quadratic edge count

Subquadratic edge countKernels for Dominating

Set

Linear edge countMeta-theorems

Graph classes

H-minor-freeH-minor-free

Ki,j-subgraph-freeKi,j-subgraph-free

bounded-genusbounded-genusbounded-genusbounded-genus

planarplanar unit-diskunit-disk

diskdisk

generalgeneral

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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)

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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

)(2

2nn

m

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TRIANGLE PACKING AND H-MATCHING

Structure theory and kernels

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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

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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

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Kernelization algorithm

Greedily build a maximal triangle packing Suppose the greedy packing contains k* copies

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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⌉ sectors (Pigeonhole Principle)

v

14v

Neighbors in each sector form a clique

Assume every disk has radius ½ If v has a neighbor x then distance |xv| ≤ 1

v

x

y

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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

v

x

y

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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 W1. Every vertex in W must be adjacent to vertex in S2. Every vertex in S has at most 12 neighbors in W

So |W| ≤ 12 |S| ≤ 12(3 k*) ∈ O(k*)

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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

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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

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RED/BLUE DOMINATING SET

Structure theory and kernels

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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|)

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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

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Reduction Rules

1. Red vertices r1, r2 such that N(r1) ∩ B ⊆ N(r2) ∩ B Delete r1

2. Blue vertices b1, b2 such that N(b1) ∩ R ⊆ N(b2) ∩ R Delete b2

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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

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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 ½

∙ ∙

radius 1

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∙

∙

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)

∙

∙

∙

∙

∙

∙∙

∙

∙

∙

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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)

Balance in colored disk graphs

AB

C

[A,B] [B,A][B,A,C][A,B,C]

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[B,A,C][A,B,C]

Balance in colored disk graphs

[B,C,A]

AB

C

[C,B,A][C,A,B]

[A,C,B]

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

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[B,A,C][A,B,C]

Balance in colored disk graphs

[B,C,A]

[C,B,A][C,A,B]

[A,C,B]

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)

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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)

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CONNECTED VERTEX COVER

Structure theory and kernels

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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)

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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 rules1. Unmarked vertex v with degree > k: mark v2. Distinct dead vertices u,v such that N(u) ⊆ N(v): delete u

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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

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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