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Algorithms 2018, 11(4), 45; https://doi.org/10.3390/a11040045

# Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems

1
Institut für Informatik I, Universität Bonn, D-53117 Bonn, Germany
2
Department of Computer Science, Lund University, 221 00 Lund, Sweden
*
Author to whom correspondence should be addressed.
Received: 6 March 2018 / Revised: 3 April 2018 / Accepted: 10 April 2018 / Published: 11 April 2018
(This article belongs to the Special Issue Algorithms for Hard Problems: Approximation and Parameterization)

# Abstract

Let R denote a connected region inside a simple polygon, P. By building barriers (typically straight-line segments) in $P \ R$ , we want to separate from R part(s) of P of maximum area. All edges of the boundary of P are assumed to be already constructed or natural barriers. In this paper we introduce two versions of this problem. In the budget fence version the region R is static, and there is an upper bound on the total length of barriers we may build. In the basic geometric firefighter version we assume that R represents a fire that is spreading over P at constant speed (varying speed can also be handled). Building a barrier takes time proportional to its length, and each barrier must be completed before the fire arrives. In this paper we are assuming that barriers are chosen from a given set B that satisfies certain conditions. Even for simple cases (e.g., P is a convex polygon and B the set of all diagonals), both problems are shown to be NP-hard. Our main result is an efficient ≈11.65 approximation algorithm for the firefighter problem, where the set B of allowed barriers is any set of straight-line segments with all endpoints on the boundary of P and pairwise disjoint interiors. Since this algorithm solves a much more general problem—a hybrid of scheduling and maximum coverage—it may find wider applications. We also provide a polynomial-time approximation scheme for the budget fence problem, for the case where barriers chosen from a set of straight-line cuts of the polygon must not cross. View Full-Text
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

MDPI and ACS Style

Klein, R.; Levcopoulos, C.; Lingas, A. Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems. Algorithms 2018, 11, 45.

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