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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
  • We will start with simple greedy-type algorithm

0 1 2 3 4
5 6 7 8
11
Independent set
  • We will start with simple greedy-type algorithm

0 1 2 3 4
5 6 7 8
12
Independent set
  • We will start with simple greedy-type algorithm

0 1 2 3 4
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

0 1 2 3 4
5 6 7 8
17
Independent set
  • We start by simple intuition

0 1 2 3 4
5 6 7 8
18
Independent set
  • We start by simple intuition

0 1 2 3 4
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

0 1 2 3 4
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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