Let B be a set of n axis-parallel boxes in d-dimensions such that each box has a corner at the origin and the other corner in the positive quadrant, and let k be a positive integer. We study the problem of selecting k boxes in B that maximize the volume of the union of the selected boxes. The research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known algorithms in any dimension d>2 enumerate all size-k subsets. We show that: * The problem is NP-hard already in 3 dimensions. * In 3 dimensions, we break the enumeration of all size-k subsets, by providing an n^O(sqrt(k)) algorithm. * For any constant dimension d, we give an efficient polynomial-time approximation scheme.
* Klee's Measure Problem To compute the volume of the union of @math (not necessarily anchored) axis-aligned boxes in @math is known as Klee's measure problem. The fastest known algorithm takes time In @math -notation, we always assume @math to be a constant, and @math is to be understood as @math . @math , which can be improved to @math if all boxes are cubes @cite_8 . By a simple reduction @cite_5 , the same running time as on cubes can be obtained on anchored boxes, which can be improved to @math for @math @cite_30 . These results are relevant to this paper because Klee's measure problem on anchored boxes (spanned by the points in @math ) is a special case of (by calling @math ).
@cite_30 @cite_5 @cite_8
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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 k -Grounded (where the projection onto the first k dimensions is a Hypervolume instance). We present a new algorithm for a classic problem in computational geometry, Klee's measure problem: given a set of n axis-parallel boxes in d-dimensional space, compute the volume of the union of the boxes. The algorithm runs in O(nd 2) time for any constant d ≥ 3. Although it improves the previous best algorithm by “just” an iterated logarithmic factor, the real surprise lies in the simplicity of the new algorithm. We also show that it is theoretically possible to beat the O(nd 2) time bound by logarithmic factors for integer input in the word RAM model, and for other variants of the problem. With additional work, we obtain an O(nd 3 polylog n)-time algorithm for the important special case of orthants or unit hypercubes (which include the so-called “hypervolume indicator problem”), and an O(n(d+1) 3 polylog n)-time algorithm for the case of arbitrary hypercubes or fat boxes, improving a previous O(n(d+2) 3)-time algorithm by Bringmann. Given n axis-parallel boxes in a fixed dimension d>=3, how efficiently can we compute the volume of the union? This standard problem in computational geometry, commonly referred to as Klee's measure problem, can be solved in time O(n^d^ ^2logn) by an algorithm of Overmars and Yap (FOCS 1988). We give the first (albeit small) improvement: our new algorithm runs in time n^d^ ^22^O^(^l^o^g^^^*^n^), where log^* denotes the iterated logarithm. For the related problem of computing the depth in an arrangement of n boxes, we further improve the time bound to near O(n^d^ ^2 log^d^ ^2^-^1n), ignoring loglogn factors. Other applications and lower-bound possibilities are discussed. The ideas behind the improved algorithms are simple. We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies, we give a fast FPRAS for all objects where one can (1) test whether a given point lies inside the object, (2) sample a point uniformly, and (3) calculate the volume of the object in polynomial time. It suffices to be able to answer all three questions approximately. We show that this holds for a large class of objects. It implies that Klee's measure problem can be approximated efficiently even though it is #P-hard and hence cannot be solved exactly in polynomial time in the number of dimensions unless P=NP. Our algorithm also allows to efficiently approximate the volume of the union of convex bodies given by weak membership oracles. For the analogous problem of the intersection of high-dimensional geometric objects we prove #P-hardness for boxes and show that there is no multiplicative polynomial-time 2^d^^^1^^^-^^^@e-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time @e-approximation.