Inapproximability of finding maximum hidden sets on polygons and terrains

摘要

How many people can hide in a given terrain, without any two of them seeing each other? We are interested in finding the precise number and an optimal placement of people to be hidden, given a terrain with n vertices. In this paper, we show that this is not at all easy: The problem of placing a maximum number of hiding people is almost as hard to approximate as the Maximum Clique problem, i.e., it cannot be approximated by any polynomial-time algorithm with an approximation ratio of n~ε for some ε > 0, unless P = NP. This is already true for a simple polygon with holes (instead of a terrain). If we do not allow holes in the polygon, we show that there is a constant ε > 0 such that the problem cannot be approximated with an approximation ratio of 1 + ε.