Approximation algorithms for geometric intersection graphs Outline v

Approximation algorithms for geometric intersection graphs

Outline v Definitions v Problem description v Techniques v. Shifting strategy

Definitions v Intersection graph v. Given a set of objects on the plane v. Each object is represented by a vertex v. There is an edge between two vertices if the corresponding objects intersect v. It can be extended to n-dimensional space v. Applications [4] v. Wireless networks (frequency assignment problems) v. Map labeling v……

Map labeling

Definitions v Intersection graphs (cont. ) v. Examples: Geometric representation Intersection graph

Definitions v ρ-approximation algorithm for optimization problems v. Runs in polynomial time v. Approximation ratio ρ v. Min: Approx/OPT ≤ ρ v. Max: OPT/Approx ≤ ρ v PTAS: Polynomial Time Approximation Scheme v. Is a class of approximation algorithms vρ = 1 + ε for every constant ε > 0

Problem description v. A unit disk graph is the intersection graph of a set of unit disks in the plane. v. We present polynomial-time approximation schemes (PTAS) for the maximum independent set problem (selecting disjoint disks). v. 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 v Maximum Independent Set for disk graphs v. Given a set S of disks on the plane, find a subset IS of S such that for any two disks D 1, D 2 IS, are disjoint v|IS| is maximized. v 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) PTAS Running time Ratio ρ v Max-Independent Set v. Unit disk graph (UDG): v. Weighted disk graph (WDG): n. O(k) 2) O(k n 1/(1 -2/k) 1/(1 -1/k)2 n. O(k ) 2) O(k n 2 (1+1/k)2 1+6/k n. O(k 3) ? ? (1+1/k)2 ? ? v Min-Vertex Cover v. UDG: v. WDG: v Min-Dominating Set v. UDG: v. 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 K 1: the squares of OPT on even lines. K 2: the squares of OPT on odd lines. OPT= k 1+k 2

Shifting strategy v Ideas: v. Partition the plane using vertical and horizontal equally separated lines v. Number vertical lines from bottom to top with 0, 1, … v. 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 v Example for unit disk graph: k = 3 0 1 2 3 4 5 6 7 8

Shifting strategy v Example

Shifting strategy v We can solve each strip independently. v Let assume we can solve each strip. v Let Ai be the value of the solution of shift i. v Let OPT denote the optimal solution. v Let OPTi be the disks of OPT intersecting active lines in shift i. v OPT = OPT 1+ OPT 2+ …+OPTk

Shifting strategy v Example

Shifting strategy v 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) v We left with disjoint squares of side length k v One square can contain at most O(k 2) disks.

Shifting strategy v The Cardinality of the solution output is at least (1 – 2 / k ) OPT v Each disk intersects only one horizontal line and one vertical line. v 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 v The set Di, j still contains an independent set of size at most (1 – 2 / k ) OPT.

Shifting strategy v Our algorithm computes a maximum independent set in each Di, j the largest such set must have cardinality at least (1 – 2 / k ) OPT ┌ ┐ v For given ε > 0 we choose k = 2/ ε to obtain (1 – ε ) OPT 2) O(k v The running time is |D|

Problem description v Min-Dominating Set for disk graphs v. Given a set S of disks on the plane, find a subset DS of S such that for any disk D S, v. D is either in DS, or v. D is adjacent to some disk in DS. v|DS| is minimized. v 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 v [1] B. S. Baker, Approximation algorithms for NP-complete Problems on Planar Graphs, J. ACM, Vol. 41, No. 1, 1994, pp. 153 -180 v [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 v [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. v [4] http: //www. tik. ee. ethz. ch/~erlebach/chorin 02 slides. pdf