On the Complexity of Nonconvex Covering

We study the problem of covering given points in Euclidean space with a minimum number of nonconvex objects of a given type. We concentrate on the one-dimensional case of this problem, whose computational complexity was previously unknown. We define a natural measure for the “degree of nonconvexity” of a nonconvex object. Our results show that for any fixed bound on the degree of nonconvexity of the covering objects the one-dimensional nonconvex covering problem can be solved in polynomial time. On the other hand without such bound on the degree of nonconvexity the one-dimensional nonconvex covering problem is NP-complete. We also consider the capacitated version of the nonconvex covering problem and we exhibit a useful property of minimum coverings by objects whose degree of nonconvexity is low.