Approximation algorithms for the geometric firefighter and budget fence problems (Q2331446)

Summary: Let \(R\) denote a connected region inside a simple polygon, \(P\). By building barriers (typically straight-line segments) in \(P \backslash R\), 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 \textit{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 \textit{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 \(\approx 11.65\) 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.