Budgeted coverage of a maximum part of a polygonal area
Stathis Zachos
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Abstract
Given is a polygonal area and the goal is to cover the polygon's interior or boundary by placing a number of stations in the interior or on the boundary. There are several parameters for this problem: what part of the polygon (interior, boundary) is to be covered, and where can stations be placed (vertices, edges, interior points); Furthermore there are minimization problems (minimum number of stations) or maximization problems (given the number of stations, maximize the covered portion of the polygon). Most of these problems are NP-hard and some are even APX-hard. We consider here a new more realistic version: the boundary is subdivided into segments with lengths and costs and the meaning of covering is relaxed and budgeted. More specifically we consider maximization problems in polygons with weighted edge segments and placement costs. We introduce a new concept: each candidate place for a station has a cost. We study the following two problems: Given a polygon with or without holes and a budget B, place stations so that the total cost of stations does not exceed B and a) the length of the boundary covered is maximized or b) the total weight of segments watched or overseen is maximized. We present constant ratio approximation algorithms for all the above problems.
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