Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems

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.