On Some City Guarding Problems

摘要

We consider guarding a city of k vertical buildings, each having a rectangular base, by placing guards only at vertices. The aim is to use the smallest number of guards. The problem is a 2.5D variant of the traditional art gallery problem, and finds applications in urban security. We give upper and lower bounds on the number of guards needed for a few versions of the problem. Specifically, we prove that [2(k-1)/3)]+1 guards are always sufficient and sometimes necessary to guard all roofs, and 1+k+[k/2] guards are always sufficient to guard the roofs, walls, and the ground, while each roof has at least one guard on it.