VC-Dimension of Visibility on Terrains

A guarding problem can naturally be modeled as a set system (U,S) in which the universe U of elements is the set of points we need to guard and our collection S of sets contains, for each potential guard g, the set of points from U seen by g. We prove bounds on the maximum VC-dimension of set systems associated with guarding both 1.5D terrains (monotone chains) and 2.5D terrains (polygonal terrains). We prove that for monotone chains, the maximum VC-dimension is 4 and that for polygonal terrains, the maximum VC-dimension is unbounded. Terrain Guarding A 1.5D (resp. 2.5D) terrain is a continuous piecewise linear univariate (resp. bivariate) function. In other words, a 1.5D terrain is a simple polygonal chain that intersects any vertical line at at most one point and a 2.5D terrain is a polygonal mesh with no holes that intersects any vertical line at at most one point. On a terrain T, either 1.5- or 2.5-dimensional, we say that two points see each other if the line segment between them does not pass under T. To guard T optimally we must find a minimum set G T of points on the terrain such that every point on T is seen by a point in G. Guarding 1.5D terrains is not known to be NP-hard but no polynomial-time exact algorithm has been found. The best polynomial-time algorithm found so far is a 5approximation algorithm 1 [10]. Guarding 2.5D terrains is NP-complete, as proved by Cole and Sharir [4]. Set Cover and VC-Dimension Set Cover is a wellstudied NP-complete optimization problem. Given a universe U of elements and a collection S of subsets of U, Set Cover asks for the minimum subset C of S such that S S2C S = U. In other words, we want to cover all