Bringing Order to Special Cases of Klee’s Measure Problem

Abstract

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 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 kGrounded (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 O(n d/2−ε )) this reduction shows that there is no algorithm with runtime O(n d/2 /2−ε ) 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].

Diversity Assessment in Many-Objective Optimization

“…DCI [55] also employs a grid environment to assess both spread and uniformity, so the number of grids needs to be pre-defined. Hypervolume calculates the volume that the obtained solution set dominates respect to a reference point [70], but it cannot be applied to MaOPs in practice due to its prohibitively high computational complexity [27]. IGD is the average distance from a reference set (samplings on the true PF) to the obtained set.…”

“…In contrast, hypervolume evaluates both convergence and diversity [22], thus many hypervolume-based MOEAs [23], [24], [25] have been developed. Although the computational complexity for calculating the exact hypervolume has been lowered [26], [27], MOEAs rely on on-line hypervolume calculation have not been applied to MaOPs [28]. R2 [29] evaluates both convergence and diversity and R2-based MOEAs for MaOPs have been reported in [30], [31].…”

Klee's Measure Problem Made Easy

“…As observed by Bringmann [9], the measure problem for general boxes in IR d can be reduced to the measure problem for orthants in IR 2d ; thus, the problem for orthants or hypercubes remains…”

Parameterized average-case complexity of the hypervolume indicator

“…As there is a parameterized reduction from KMP to HYP [10], also HYP is W[1]-hard. This implies that there is no algorithm for HYP with runtime f (d) poly(n) for any function f unless FPT = W [1].…”

“…It was recently shown that any algorithm for KMP on (unit) cubes yields an algorithm for HYP with the same asymptotic runtime [10]. This implies that the algorithm of Bringmann [9] for KMP on cubes also gives an algorithm with runtime O(n (d+2)/3 ) for HYP.…”