Approximation algorithms for geometric intersection graphs

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Approximation algorithms for geometric
intersection graphs
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Outline

• Definitions
• Problem description
• Techniques
• Shifting strategy

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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

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Map labeling
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Definitions

• Intersection graphs (cont.)
• Examples

Geometric representation
Intersection graph
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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

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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.

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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.

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Independent Set
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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

• 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
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What known? (Using shifting strategy)
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• 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 ??
  ??

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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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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 k1k2
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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.

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Shifting strategy

• Example for unit disk graph k 3

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Shifting strategy

• Example

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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
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

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Shifting strategy

• Example

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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.

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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.

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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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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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