Given a set B of d-dimensional boxes (i.e., axis-aligned hyperrectangles), a minimum coverage kernel is a subset of B of minimum size covering the same region as B. Computing it is NP-hard, but as for many similar NP-hard problems (e.g., Box Cover, and Orthogonal Polygon Covering), the problem becomes solvable in polynomial time under restrictions on B. We show that computing minimum coverage kernels remains NP-hard even when restricting the graph induced by the input to a highly constrained class of graphs. Alternatively, we present two polynomial-time approximation algorithms for this problem: one deterministic with an approximation ratio within 0(logn), and one randomized with an improved approximation ratio within 0(lgOPT)(with high probability).
