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Minimum Vertex Cover in Rectangle Graphs

R. Bar-Yehuda, D. Hermelin, and D. Rawitz

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Vertex Cover in Rectangle Graphs

• R’ R s.t. R - R’ is pairwise non-intersecting:

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Vertex Cover in Rectangle Graphs

• R’ R s.t. R - R’ is pairwise non-intersecting:

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Vertex Cover in Rectangle Graphs

• R’ R s.t. R - R’ is pairwise non-intersecting:

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Applications

• Critical section scheduling• rectangles are jobs

• x-axis = time • y-axis = shared resource

• dump min number of jobs to get a feasible schedule

• Automated map labeling• rectangles are labels• dump min number of labels to get a pairwise

non-intersecting labeling

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Previous Work

• NP-hard [Fowler et al. 1981]• PTAS for equal height rectangles [Agarwal et al. 1998]• PTAS for squares [Chan 2003]• PTAS for bounded aspect-ratio rectangles [Erlebach et al. 2005]• EPTAS for unit-squares [Marx 2005]

• f(1/) nO(1) instead of nf(1/). • EPTAS for squares [van Leeuwen 2006]• PTAS for non-crossing rectangles [Chan and Har-Peled 2009]

• no pair of rectangles intersect as follows:

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Our results

1. EPTAS for non-crossing rectangles• Improving [Chan and Har-Peled 2009] from

a PTAS to an EPTAS.• Extends to pseudo-disks.

2. (1.5 + )-approx. for general rectangles• First approximation for general rectangles

which is better than 2. • Extends to the weighted case.

Non-Crossing RectanglesArrangement graph AR of R:• V(AR) := intersection points (joints)• E(AR) := rectangle edges

Our EPTAS is based on Baker's scheme and uses the planarity of the arrangement graph

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k-outerplanar graphs • Label vertices of a plane graph

by level.• All vertices on exterior face

level 1.• All vertices on exterior face

when vertices of levels 1 … i are removed are on level i+1.

• Graph is k-outerplanar when at most k levels.

• Theorem: k-outerplanar graphs have treewidth at most 3k – 1. 3-outerplanar

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Vertex cover on k-outerplanar graphs

• For fixed k, finding a minimum Vertex Cover in a k-outerplanar graph can be solved in linear time (O(n*23k) time).– Dynamic programming on the 3k-tree width

decomposition

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Tree decomposition• A tree decomposition:

– Tree with a vertex set called bag associated to every node.

– For all edges {v,w}: there is a set containing both v and w.

– For every v: the nodes that contain v form a connected subtree.

b c

d ef

a abc

c

c de

h

hff

ag

g

a g

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Tree decomposition• A tree decomposition:

– Tree with a vertex set called bag associated to every node.

– For all edges {v,w}: there is a set containing both v and w.

– For every v: the nodes that contain v form a connected subtree.

b c

d ef

a abc

c

c de

h

hff

ag

g

a g

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Tree decomposition• A tree decomposition:

– Tree with a vertex set called bag associated to every node.

– For all edges {v,w}: there is a set containing both v and w.

– For every v: the nodes that contain v form a connected subtree.

b c

d ef

a abc

c

c de

h

hff

ag

g

a g

Treewidth of graph G:

1||max iIi X

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Using modified Baker’s scheme• For each i in {1,2, …, k} do– Vi= all vertices in levels i, i+k, i+2k, i+3k, …– V-Vi is (k-1)-outerplanar.– Thus V-Vi has 3(k-1)-tree width– Ri=R(Vi )={rR : r contains a point in Vi}– Thus R-Ri=R-R(Vi ) has 6(k-1)-tree width

• Let t=arcmini{|Ri|}• Construct decomposition tree of R-Rt From the V-Vt tree• Using DP on the tree find minimum VC of R-Rt

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Approximation ratio

{1,2,. .. , }

{ }i k

i iSolution Min Opt R

t tSolution Opt R

{1,2,..., } use LRi k

iRc ROPT OPT

k k

use Nemhuser&Trot2 erc OPTOPTk

2(1 )c OPTk

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Open Problems

1. EPTAS for weighted non-crossing rectangles?

2. PTAS for general rectangles? Hardness?

3. Constant approx. for independent set?

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