# Art Gallery Problem for Monotone Polygons.

December 2013

## Date of Award

Winter 12-12-2014

## 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)

Art Gallery problem is one of the well known problems in Computational Geometry. Given a polygon P in R2 , we are asked to find the minimum number of points interior of P to guard the entire polygon P. Our study is on a restricted version of the problem, where the given polygon P is monotone with respect to x-axis. Here also different variations may be studied, namely point guarding, vertex guarding and edge guarding. Vertex Guarding deals with the case when guards are placed only at vertices of the polygon. Edge guarding deals with the case when guards are placed only at the boundary of the polygon. Point guarding deals with the case when guards can be placed anywhere inside the polygon. It is a restricted version of the Set Cover problem which is known to be NP Complete and can not be approximated to a constant approximation factor unless P = NP. For any simple polygon, point guarding problem can be formulated to set cover as given in [15]. Initially Chen et al[6] proved vertex guarding to be NP Hard. But, their proof is still omitted and is under verification. After that, Erik Krohn and Bengt J. Nillson has proved its vertex guarding of monotone polygon to be NP Hard in [2]. But, its interior guarding does not immediately follow from that claim. The same authors Erik Krohn and B. J. Nillson[7] have proved its interior guarding version to be NP Hard. It has a related problem which deals with guarding a terrain. Guarding a terrain is also NP Hard. Erik Krohn and James King [8] gave a proof of that. About guarding interior of a polygon, we know some basic results[1] that n/3 guards are always sufficient and occasionally necessary to guard a polygon. In Chapter 3, we give an approximation algorithm to guard a terrain and briefly describe other works related to it referring them. In Chapter 4, we have discussed interior guarding of monotone polygon. First we describe that a monotone polygon can be guarded with minimum number of guards when the polygon is ymonotone and also axis parallel(also called as horizontally convex). Also, we have discussed a constant factor approximation algorithm given by Bengt J. Nillson[5] when the polygon is x-monotone. It provides approximation factor 12. After that we propose an algorithm for a special sub-case when the polygon is x-monotone and also their two extreme points are mutually visible to each other. In that algorithm, we have conjectured that it is expected to give a 4-factor approximation algorithm. We have given a brief informal justification why it should give 4-factor approximation algorithm.In this thesis, we have reviewed these following works in detail. These are 4- factor approximation algorithm for terrain guarding problem[10], algorithm to guard monotone orthogonal polygon[14] and constant factor algorithm for monotone polygon given by Bengt J. Nillson[5]. And finally we have provided our approach in the following sections of Chapter 4. In Section 4.1, we have given an approach where the input polygon in uni-monotone and in Section 4.4, for a special sub-case of the x-monotone polygon.

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:28843293

## Control Number

ISI-DISS-2013-281