Bringing Order to Special Cases of Klee’s Measure Problem

Abstract

Klee’s Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in ℝ^d. Omitting logarithmic factors, the best algorithm has runtime $\mathcal{O}^*(n^{d/2})$ [Overmars,Yap’91]. There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex), and k -Grounded (where the projection onto the first k dimensions is a Hypervolume instance).

In this paper we bring some order to these special cases by providing reductions among them. In addition to the trivial inclusions, we establish Hypervolume as the easiest of these special cases, and show that the runtimes of Unitcube-KMP and Cube-KMP are polynomially related. More importantly, we show that any algorithm for one of the special cases with runtime T(n,d) implies an algorithm for the general case with runtime T(n,2d), yielding the first non-trivial relation between KMP and its special cases. This allows to transfer W[1]-hardness of KMP to all special cases, proving that no n ^o(d) algorithm exists for any of the special cases assuming the Exponential Time Hypothesis. Furthermore, assuming that there is no improved algorithm for the general case of KMP (no algorithm with runtime $\mathcal{O}(n^{d/2 - \epsilon})$) this reduction shows that there is no algorithm with runtime $\mathcal{O}(n^{\lfloor d/2 \rfloor /2 - \epsilon})$ for any of the special cases. Under the same assumption we show a tight lower bound for a recent algorithm for 2-Grounded[Yıldız,Suri’12].