Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems
Institut für Informatik I, Universität Bonn, D-53117 Bonn, Germany
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
denote a connected region inside a simple polygon, P
. By building barriers (typically straight-line segments) in
, 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.
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MDPI and ACS Style
Klein, R.; Levcopoulos, C.; Lingas, A. Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems. Algorithms 2018, 11, 45.
Klein R, Levcopoulos C, Lingas A. Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems. Algorithms. 2018; 11(4):45.
Klein, Rolf; Levcopoulos, Christos; Lingas, Andrzej. 2018. "Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems." Algorithms 11, no. 4: 45.
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