LATIN 2014: LATIN 2014: Theoretical Informatics pp 261-272

# Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems

• Rolf Klein
• Christos Levcopoulos
• Andrzej Lingas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

## Abstract

Let R denote a connected region inside a simple polygon, P. By building 1-dimensional barriers in P ∖ R, we want to separate from R part(s) of P of maximum area. 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 a certain linearity condition. For example, this condition is satisfied for barrier curves in general position, if any two barriers cross at most once.

Even for simple cases (e. g., P 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. Since this algorithm solves a much more general problem—a hybrid of scheduling and maximum coverage—it may find wider application. We also provide a polynomial-time approximation scheme for the budget fence problem, for the case where barriers chosen from B must not cross.

## Keywords

Approximation Algorithm Convex Polygon Simple Polygon Constant Approxi Algebraic Degree
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Rolf Klein
• 1
• Christos Levcopoulos
• 2
• Andrzej Lingas
• 2
1. 1.Institut für Informatik IUniversität BonnBonnGermany
2. 2.Department of Computer ScienceLund UniversityLundSweden