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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 ρ-approximation algorithm for optimization problems Runs in polynomial time Approximation ratio ρ Min: Approx/OPT ρ Max: OPT/Approx ρ PTAS: Polynomial Time Approximation Scheme Is a class of approximation algorithms ρ = 1 + ε for every constant ε > 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 0 1 2 3 4 5 6 7 8

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

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

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Independent set Can we improve the greedy algorithm? 0 1 2 3 4 5 6 7 8

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Do we need the representation

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What known? (Using shifting strategy) Max-Independent Set Unit disk graph (UDG): n O(k) 1/ ( 1-2/k ) Weighted disk graph (WDG): n O(k 2 ) 1/(1-1/k) 2 Min-Vertex Cover UDG: n O(k 2 ) (1+1/k) 2 WDG: n O(k 2 ) 1+6/k Min-Dominating Set UDG: n O(k 3 ) (1+1/k) 2 WDG: ?? ?? Running timeRatio PTAS ρ

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

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

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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

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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 0 1 2 3 4 5 6 7 8

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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 A i be the value of the solution of shift i. Let OPT denote the optimal solution. Let OPT i be the disks of OPT intersecting active lines in shift i. OPT = OPT 1 + OPT 2 + …+OPT k

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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 < k Let D i,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(k 2 ) 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 D i,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 D i,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(k 2 )

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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. 238–274. [4] http://www.tik.ee.ethz.ch/~erlebach/chorin02slides.pdf

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