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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 jansen polynomial kernels for hard problems on disk graphs

Bart JansenPolynomial Kernels for Hard Problems on Disk Graphs

Accepted for presentation at SWAT 2010


Overview
Overview

  • Introduction

    • Kernelization

    • Graph classes

  • Kernels

    • Triangle Packing, Kt-matching, H-matching

    • Red/Blue Dominating Set

    • Connected Vertex Cover

  • Conclusion


Kernelization for graph problems
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


Recent kernelization results
Recent kernelization results

Bad news

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


Expanding the range of good news
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


Graph classes
Graph classes

Linear edge count

Meta-theorems

Quadratic edge count

planar

unit-disk

bounded-genus

bounded-genus

H-minor-free

disk

Our kernels

Ki,j-subgraph-free

Subquadratic edge count

Kernels for Dominating Set

general


Disk graphs
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)


Properties of disk graphs
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


Triangle packing and h matching

Structure theory and kernels

Triangle packing and h-matching


Triangle packing
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


Triangle packing1
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


Kernelization algorithm
Kernelization algorithm

  • Greedily build a maximal triangle packing

  • Suppose the greedy packing contains k* copies


Neighborhood clique lemma
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


Neighbors in each sector form a clique
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


Neighbors in each sector form a clique1
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


Analysis of kernel size
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*)


Extension to k t matching
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


Extension to h matching
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


Red blue dominating set

Structure theory and kernels

Red/blue dominating set


Red blue dominating set1
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|)


Background
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


Reduction rules
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


Balance
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


Balance in colored unit disk graphs
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


Proof
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)


Balance in colored disk graphs
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]

[B,A,C]

A

B

C


Balance in colored disk graphs1
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

[B,C,A]

[A,C,B]

C

[C,A,B]

[C,B,A]


Balance in colored disk graphs2
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]


Summary of kernels for red blue dominating set
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)


Connected vertex cover

Structure theory and kernels

Connected vertex cover


Connected vertex cover1
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)


Annotated connected vertex cover
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


Analysis
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


Conclusion and discussion
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


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