[Paper Review] Maximum Volume Subset Selection for Anchored Boxes

Karl Bringmann, Sergio Cabello+1 more|arXiv (Cornell University)|Jan 1, 2017

Computational Geometry and Mesh Generation19 citations

TL;DR

This paper presents an n^O(√k) algorithm for the maximum volume subset selection problem in 3D anchored boxes, breaking the previous Ω(n choose k) lower bound. It also introduces an efficient polynomial-time approximation scheme (EPTAS) for any constant dimension d, significantly improving upon the prior (1−1/e)-approximation. The work establishes NP-hardness in 3D and provides a novel approach combining planar separators and dynamic programming with rounding techniques for approximation.

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 $Ω\big(\binom{n}{k}\big)$. We show that: - The problem is NP-hard already in 3 dimensions. - In 3 dimensions, we break the bound $Ω\big(\binom{n}{k}\big)$, by providing an $n^{O(\sqrt{k})}$ algorithm. - For any constant dimension $d$, we present an efficient polynomial-time approximation scheme.

Motivation & Objective

• To address the computational complexity of selecting k anchored boxes in d-dimensional space to maximize the union volume.
• To close the gap between the best-known algorithms and the theoretical lower bound in 3D, where previous algorithms required Ω(n choose k) time.
• To develop a polynomial-time approximation scheme (EPTAS) for any constant dimension d, improving upon the prior (1−1/e)-approximation.
• To establish NP-hardness of the problem in 3D, showing it is computationally intractable even in low dimensions.

Proposed method

• Proved NP-hardness in 3D via a reduction from independent set on planar graphs of maximum degree 3, embedding the problem into 3D using two parallel planes.
• Designed an n^O(√k) algorithm for 3D by projecting the 3D problem into 2D and applying planar separator techniques to partition and solve subproblems.
• Introduced a (1−ε)-approximation algorithm using a grid-based rounding scheme and dynamic programming over grid cells to manage the exponential dependence on ε and d.
• Applied a multiple-choice 0/1 knapsack formulation to efficiently combine solutions from grid cells, leveraging known PTAS techniques for this subproblem.
• Used coordinate rounding and offset iteration to ensure approximation guarantees, with error bounded by O(ε) via geometric and combinatorial arguments.
• Traced back dynamic programming tables to reconstruct a feasible solution S ⊆ P of size at most k, achieving a (1−O(ε))-approximation to the optimal volume.

Experimental results

Research questions

• RQ1 Is the maximum volume subset selection problem NP-hard in 3 dimensions, despite being solvable in polynomial time in 2D?
• RQ2 Can the Ω(n choose k) time bound for 3D volume selection be broken, and if so, by how much?
• RQ3 Is there an efficient polynomial-time approximation scheme (EPTAS) for the problem in any constant dimension d ≥3?
• RQ4 What is the optimal trade-off between approximation quality (ε), running time, and dependence on k and d?

Key findings

• The maximum volume subset selection problem is NP-hard in 3 dimensions, even when d is a constant, resolving a long-standing open question.
• An n^O(√k) algorithm is presented for 3D volume selection, breaking the previous Ω(n choose k) lower bound and significantly improving the state of the art.
• For any constant dimension d, the paper provides an EPTAS running in time O(n · ε^−d(log n + k + 2^O(ε^−2 log 1/ε)^d)), which is polynomial in n and k for fixed d and ε.
• The algorithm achieves a (1−O(ε))-approximation to the optimal volume, with the approximation error controlled by ε and independent of n.
• The running time of the improved EPTAS is bounded by O(2^O(ε^−2 log 1/ε)^d · n log n), which is efficient for fixed d and ε.
• The solution construction via dynamic programming and backtracking ensures a feasible subset of size at most k that achieves the desired approximation ratio.