Approximation algorithms
for geometric intersection
graphs
Outline
Definitions
Problem description
Techniques
Shifting strategy
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
……
Map labeling
Definitions
Intersection graphs (cont.)
Examples:
Geometric
representation
Intersection
graph
Definitions
ρ-approximation algorithm for optimization
problems
Runs in polynomial time
Approximation ratio ρ
Min: Approx/OPT ≤ ρ
Max: OPT/Approx ≤ ρ
PTAS: Polynomial Time Approximation Scheme
Is a class of approximation algorithms
ρ = 1 + ε for every constant ε > 0
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.
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.
Independent Set
Independent set
We will start with simple greedy-type algorithm
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Independent set
We will start with simple greedy-type algorithm
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Independent set
We will start with simple greedy-type algorithm
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Independent set
Can we improve the greedy algorithm?
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Do we need the representation
What known? (Using shifting
strategy)
PTAS
Running time
Ratio ρ
Max-Independent Set
Unit disk graph (UDG):
Weighted disk graph (WDG):
nO(k)
2)
O(k
n
1/(1-2/k)
1/(1-1/k)2
2
(1+1/k)2
1+6/k
Min-Vertex Cover
UDG:
WDG:
nO(k )
2)
O(k
n
Min-Dominating Set
UDG:
WDG:
3
nO(k )
??
(1+1/k)2
??
Independent set
We start by simple intuition
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Independent set
We start by simple intuition
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Independent set
We start by simple intuition
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K1: the squares
of OPT on even
lines.
K2: the squares
of OPT on odd
lines.
OPT= k1+k2
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.
Shifting strategy
Example for unit disk graph: k = 3
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Shifting strategy
Example
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
Shifting strategy
Example
Shifting strategy
For each pair of integers ( i , j ) such that
0 ≤ i, j < 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.
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.
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 ε > 0 we choose k = 2/ ε to obtain
(1 – ε ) OPT
2)
O(k
The running time is |D|
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.
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.
238–274.
 [4] http://www.tik.ee.ethz.ch/~erlebach/chorin02slides.pdf