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Algorithms 2018, 11(4), 45; doi: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)
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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
Keywords: budget fence problem; firefighter problem; PTAS; scheduling; set cover; time complexity budget fence problem; firefighter problem; PTAS; scheduling; set cover; time complexity
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Klein, R.; Levcopoulos, C.; Lingas, A. Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems. Algorithms 2018, 11, 45.

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