Approximation algorithms for geometric intersection

graphs

Outline

Definitions

Problem description

TechniquesShifting strategy

DefinitionsIntersection graph

Given a set of objects on the plane Each object is represented by a vertexThere is an edge between two vertices if the

corresponding objects intersectIt can be extended to n-dimensional spaceApplications [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 problemsRuns in polynomial timeApproximation ratio ρ

Min: Approx/OPT ≤ ρMax: OPT/Approx ≤ ρ

PTAS: Polynomial Time Approximation SchemeIs a class of approximation algorithmsρ = 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 graphsGiven a set S of disks on the plane, find a subset IS

of S such that for any two disks D1,D2IS, 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 SetUnit disk graph (UDG): nO(k) 1/(1-2/k)Weighted disk graph (WDG): nO(k2) 1/(1-1/k)2

Min-Vertex CoverUDG: nO(k2) (1+1/k)2

WDG: nO(k2) 1+6/k

Min-Dominating SetUDG: nO(k3) (1+1/k)2

WDG: ?? ??

Running time RatioPTAS

ρ

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= 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 = 30 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 strategyFor 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 strategyThe Cardinality of the solution output is at least

(1 – 2 / k ) OPTEach 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

The running time is |D|O(k2)

Problem description

Min-Dominating Set for disk graphsGiven a set S of disks on the plane, find a subset DS

of S such that for any disk DS,D is either in DS, orD 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