The Complexity of Combinatorial Optimization Problems on d-Dimensional Boxes

The Maximum Independent Set problem in d-box graphs, i.e., in intersection graphs of axis-parallel rectangles in Rd, is known to be NP-hard for any fixed d ≥ 2. A challenging open problem is, how close the solution can be approximated by a polynomial time algorithm. For the restricted case of d-boxes with bounded aspect ratio a PTAS exists [12]. In general case no polynomial time algorithm with approximation ratio o(logd−1 n) for a set of n d-boxes is known. In this paper we prove APX-hardness of the Maximum Independent Set problem in d-box graphs for any fixed d ≥ 3. We give an explicit lower bound 245 244 on efficient approximability for this problem unless P = NP. Additionally, we provide a generic method how to prove APX-hardness for other graph optimization problems in d-box graphs for any fixed d ≥ 3.