The Complexity of Combinatorial Optimization Problems on d‐Dimensional Boxes

Abstract

The MAXIMUM INDEPENDENT SET problem in d‐box graphs, i.e., in intersection graphs of axis‐parallel rectangles in $\mathbb{R}^d$, is known to be NP‐hard for any fixed $d\geq 2$. A challenging open problem is that of how closely the solution can be approximated by a polynomial time algorithm. For the restricted case of d‐boxes with bounded aspect ratio a PTAS exists [T. Erlebach, K. Jansen, and E. Seidel, SIAM J. Comput., 34 (2005), pp. 1302–1323]. In the general case no polynomial time algorithm with approximation ratio $o(\log^{d-1} n)$ 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 $d\geq 3$. We give an explicit lower bound $\frac{245}{244}$ on efficient approximability for this problem unless $\PP=\text{\rm NP}$. Additionally, we provide a generic method how to prove APX‐hardness for other graph optimization problems in d‐box graphs for any fixed $d\geq 3$.