# Algorithms on Geometric Graphs.

December 2010

## Date of Award

Winter 12-12-2011

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Advance Computing and Microelectronics Unit (ACMU-Kolkata)

## Supervisor

Nandy, Subhas Chandra (ACMU-Kolkata; ISI)

## Abstract (Summary of the Work)

Geometric intersection graphs are intensively studied both for their practical motivations and interesting theoretical properties. Map labelling, frequency allocation in wireless network, resource allocation in line network are some of the areas where geometric intersection graphs play an important role in formulating problems. Here two types of problems are usually considered: (i) characterization problems, and (ii) solving some useful optization problems. In the characterization problem, given an arbitrary graph, one needs to check whether it belongs to the intersection graph of a desired type of objects. The second kind of problem deals with designing efficient algorithm for solving some useful optization problems for an intersection graph of a known type of objects. It is to be noted that several practically useful optization problems, for example, finding the largest clique, minimum vertex cover, maximum independent set, etc. are NP-hard for general graph. There are some problems for which getting an efficient approximation algorithm with good approximation factor is also very difficult. In this area of research, the geometric properties of the intersecting objects are used to design efficient algorithm for these optimization problems. The characterization problem is important in the sense that for the intersection graph of some types of objects, efficient algorithms are sometimes already available for solving the desired optimization problem.The simplest type of geometric intersection graph is the interval graph, which is obtained by the overlapping information of a set of intervals on a real line. The characterization problem can be easily solved in O(|V | + |E|) time by showing that the graph is chordal and its complementary graph is a comparability graph [Gol04].All the standard graph-theoretic optimization probelms, for example, finding minimu vertex cover, maximum independent set, largest clique, minimum clique cover, minimum coloring, etc, can be solved in polynomial time for the interval graph [Gol04]. Any graph G = (V, E) can be represented as the intersection graph of a set of axis parallel boxes in some dimension. The boxicity of a graph with n nodes is the minimum dimension d such that the given graph can be represented as an intersection graph of n axis parallel boxes in dimension d. A graph has boxicity at most one if and only if it as an interval graph. Many optimization problems can be solved or approximated more efficiently for graphs with bounded boxicity. For instance, the maximum clique problem for the intersectio graph of axis parallelrectangles in 2D can be computed in O(n log n) time using a plane sweep strategy [NB95].The maximum independent set of rectangle intersection graph is extensively used in map labelling. The maximum independent set for equal height rectangle intersection graph are shown to admit a PTAS. A 2-factor approximation algorithm is very easy to get in O(n log n) time [AvKS98]. In Chapter 4 we propose that piercing set for bounded height rectangles is fixed parameter tractable.A graph G = (V, E) is said to be a disk graph if it is obtained from the intersection of a set of disks. Unit disk graphs play important role in formulating different important problems in mobile ad hoc network. In mobile network, the base stations can be viewed as nodes on unit disk graph; the range of each base station is the same. Different problems on this network can be formulated as the graph-theoretic problems on unit disk graph. Recognizing whether an arbitrary graph is unit disk graph is NP-cmplete [BK98]. Maximum clique can be computed in polynomial time for unit disk graph[CCJ90].In Chapter 3 we propose a PTAS for maximum independent set of unit disk graph. A 3-factor approximation algorithm for minimum clique cover of unit disk graph is also described in that chapter. We also propose a 4-factor approximation algorithm for the minimum piercing set of points for a set of unit disks distributed randomly on the plane. Here the piercing points can be chosen to be any point on the plane. In the discrete piercing set problem, a point set P is given. The unit circles are all centered at the points in P. The objective is to choose the minimum set of points in P to pierce all the circles. We propose a 15-factor approximation algorithm for this problem

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843205

## Control Number

ISI-DISS-2010-250

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/6407

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