On Some City Guarding Problems

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Abstract

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 \(\lfloor\frac{2(k-1)}{3}\rfloor + 1\) guards are always sufficient and sometimes necessary to guard all roofs, and \(1 + k + \lfloor \frac{k}{2}\rfloor\) guards are always sufficient to guard the roofs, walls, and the ground, while each roof has at least one guard on it.