Natural Wireless Localization is NP-hard

We consider a special class of art gallery problems inspired by wireless localization. Given a polygonal region P, place and orient guards each of which broadcasts a unique key within a fixed angular range. In contrast to the classical art gallery setting, broadcasts are not blocked by the boundary of P. At any point in the plane one must be able to tell whether or not one is located inside P only by looking at the set of keys received. We prove NP-hardness of one variant of the problem, namely the natural setting where guards may be placed aligned to a boundary edge or two consecutive boundary edges of P only. Introduction. Art gallery problems are a classic topic in discrete and computational geometry. A new direction has recently been introduced by Eppstein, Goodrich, and Sitchinava [1]. They propose to modify the concept of visibility by not considering the edges of the polygon/gallery as blocking. This changes the problem quite drastically because it breaks up a certain locality where the shape of the polygon dictates the possible placement of guards. A basic ingredient in hardness proofs for the classical setting is a small pocket/spike of the polygon which can only be guarded from a nearby point because the bounding polygon edges shield it away from the rest of the world. This argument breaks down if the edges do not block visibility. The motivation for this model stems from communication in wireless networks where the signals are not blocked by walls, either. For illustration, suppose you run a café (modeled, say, as a simple polygon P ) and you want to provide wireless Internet access to your customers. But you do not want the whole neighborhood to use your infrastructure. Instead, Internet access should be limited to those people who are located within the café. To achieve this, you can install a certain number of devices, let us call them guards, each of which broadcasts a unique (secret) key in an arbitrary but fixed angular range. The goal is to place guards and adjust their angles in such a way that everybody who is inside the café can prove this fact just by naming the keys received and nobody who is outside the café can provide such a proof. Formally this means that P can be described by a monotone Boolean formula over the keys, that is, a formula using the operators And and Or only, negation is not allowed. (See [1, 2, 3].)