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Computer Science, Computational Geometry, cs.CG,Computer Science, Data Structures and Algorithms, cs.DS
Abstract:
Let $B$ be a set of $n$ axis-parallel boxes in $\mathbb{R}^d$ such that each
box has a corner at the origin and the other corner in the positive quadrant of
$\mathbb{R}^d$, 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. This 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 running time in
any dimension $d \ge 3$ is $\Omega\big(\binom{n}{k}\big)$. We show that:
- The problem is NP-hard already in 3 dimensions.
- In 3 dimensions, we break the bound $\Omega\big(\binom{n}{k}\big)$, by
providing an $n^{O(\sqrt{k})}$ algorithm.
- For any constant dimension $d$, we present an efficient polynomial-time
approximation scheme.
