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## Approximation algorithms for geometric intersection graphs

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Title: Approximation algorithms for geometric intersection graphs

1
Approximation algorithms for geometric
intersection graphs
2
Outline
• Definitions
• Problem description
• Techniques
• Shifting strategy

3
Definitions
• Intersection graph
• Given a set of objects on the plane
• Each object is represented by a vertex
• There is an edge between two vertices if the
corresponding objects intersect
• It can be extended to n-dimensional space
• Applications 4
• Wireless networks (frequency assignment problems)
• Map labeling

4
Map labeling
5
Definitions
• Intersection graphs (cont.)
• Examples

Geometric representation
Intersection graph
6
Definitions
• ?-approximation algorithm for optimization
problems
• Runs in polynomial time
• Approximation ratio ?
• Min Approx/OPT ?
• Max OPT/Approx ?
• PTAS Polynomial Time Approximation Scheme
• Is a class of approximation algorithms
• ? 1 e for every constant e gt 0

7
Problem description
• A unit disk graph is the intersection graph of a
set of unit disks in the plane.
• We present polynomial-time approximation schemes
(PTAS) for the maximum independent set problem
(selecting disjoint disks).
• The idea is based on a recursive subdivision of
the plane. They can be extended to intersection
graphs of other disk-like geometric objects
(such as squares or regular polygons), also in
higher dimensions.

8
Independent Set
• Maximum Independent Set for disk graphs
• Given a set S of disks on the plane, find a
subset IS of S such that for any two disks
D1,D2?IS, are disjoint
• IS is maximized.
• We are given a set of unit disks and want to
compute a maximum independent set, i.e., a subset
of the given disks such that the disks in the
subset are pairwise disjoint and their
cardinality is maximized.

9
Independent Set
10
Independent set

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5 6 7 8
11
Independent set

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5 6 7 8
12
Independent set

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5 6 7 8
13
Independent set
• Can we improve the greedy algorithm?

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5 6 7 8
14
Do we need the representation
15
What known? (Using shifting strategy)
?
• Max-Independent Set
• Unit disk graph (UDG) nO(k)
1/(1-2/k)
• Weighted disk graph (WDG) nO(k2)
1/(1-1/k)2
• Min-Vertex Cover
• UDG
nO(k2) (11/k)2
• WDG
nO(k2) 16/k
• Min-Dominating Set
• UDG
nO(k3) (11/k)2
• WDG ??
??

16
Independent set
• We start by simple intuition

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5 6 7 8
17
Independent set
• We start by simple intuition

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5 6 7 8
18
Independent set
• We start by simple intuition

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5 6 7 8
K1 the squares of OPT on even lines. K2 the
squares of OPT on odd lines. OPT k1k2
19
Shifting strategy
• Ideas
• Partition the plane using vertical and horizontal
equally separated lines
• Number vertical lines from bottom to top with 0,
1,
• Given a constant k, there is a group of vertical
(horizontal) lines whose line numbers r (mod k)
and the number of disks that intersect those
lines is not larger than 1/k of total number of
disks.

20
Shifting strategy
• Example for unit disk graph k 3

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5 6 7 8
21
Shifting strategy
• Example

22
Shifting strategy
• We can solve each strip independently.
• Let assume we can solve each strip.
• Let Ai be the value of the solution of shift i.
• Let OPT denote the optimal solution.
• Let OPTi be the disks of OPT intersecting active
lines in shift i.
• OPT OPT1 OPT2 OPTk

23
Shifting strategy
• Example

24
Shifting strategy
• For each pair of integers ( i , j ) such that
• 0 i, j lt k
• Let Di,j be the subset of disks obtained by
removing all disks that intersects a vertical
line at x i kp (p is integer)
• and horizontal line at x j kp (p is
integer)
• We left with disjoint squares of side length k
• One square can contain at most O(k2) disks.

25
Shifting strategy
• The Cardinality of the solution output is at
least
• (1 2 / k ) OPT
• Each disk intersects only one horizontal line and
• one vertical line.
• There exists a value of i such that at most OPT/k
• disks in OPT intersects vertical lines x i
kp Similarly, there is a value of j such that
at most OPT/k disks in OPT intersects horizontal
lines
• x j kp
• The set Di,j still contains an independent set of
size at most (1 2 / k ) OPT.

26
Shifting strategy
• Our algorithm computes a maximum independent set
in each Di,j the largest such set must have
cardinality at least
• (1 2 / k ) OPT
• For given e gt 0 we choose k 2/ e to obtain
• (1 e ) OPT
• The running time is DO(k2)

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Problem description
• Min-Dominating Set for disk graphs
• Given a set S of disks on the plane, find a
subset DS of S such that for any disk D?S,
• D is either in DS, or
• D is adjacent to some disk in DS.
• DS is minimized.
• Whether MDS for disk graph has a PTAS or not is
still an open question. In my project, I first
assume it exists, and then try to find a PTAS
using existing techniques.

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38
References
• 1 B. S. Baker, Approximation algorithms for
NP-complete Problems on Planar Graphs, J. ACM,
Vol. 41, No. 1, 1994, pp. 153-180
• 2 T. Erlebach, K. Jansen, and E. Seidel,
Polynomial-time approximation schemes for
geometric intersection graphs, Siam J. Comput.
Vol. 34, No. 6, pp. 1302-1323
• 3 Harry B. Hunt III, M. V. Marathe, V.
Radhakrishnan, S. S. Ravi, D. J. Rosenkrantz, R.
E. Stearns, NC-approximation schemes for NP- and
PSPACE-hard problems for geometric graphs, J.
Algorithms, 26 (1998), pp. 238274.
• 4 http//www.tik.ee.ethz.ch/erlebach/chorin02sl
ides.pdf

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