Loading...

PPT – Approximation algorithms for geometric intersection graphs PowerPoint presentation | free to download - id: 6d5500-NWYwN

The Adobe Flash plugin is needed to view this content

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

- ?-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

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

0 1 2 3 4

5 6 7 8

Independent set

- We will start with simple greedy-type algorithm

0 1 2 3 4

5 6 7 8

Independent set

- We will start with simple greedy-type algorithm

0 1 2 3 4

5 6 7 8

Independent set

- Can we improve the greedy algorithm?

0 1 2 3 4

5 6 7 8

Do we need the representation

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 ??

??

Independent set

- We start by simple intuition

0 1 2 3 4

5 6 7 8

Independent set

- We start by simple intuition

0 1 2 3 4

5 6 7 8

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

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

0 1 2 3 4

5 6 7 8

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 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.

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 e gt 0 we choose k 2/ e to obtain
- (1 e ) OPT
- The running time is DO(k2)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

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.

(No Transcript)

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

(No Transcript)