# Maximum Volume Subset Selection for Anchored Boxes

## TL;DR

This paper investigates the maximum volume subset selection problem for anchored boxes in multiple dimensions, proving NP-hardness in 3D, providing a subexponential algorithm in 3D, and offering an efficient approximation scheme for fixed dimensions.

## Contribution

It establishes NP-hardness in three dimensions, introduces a subexponential algorithm for 3D, and develops an efficient approximation scheme for fixed dimensions.

## Key findings

- NP-hardness in 3D established.
- Subexponential $n^{O(\sqrt{k})}$ algorithm for 3D.
- Efficient polynomial-time approximation scheme for fixed dimensions.

## 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.