A Bicriteria Approximation Scheme for the Minimum Well-Separated Pair Decomposition

Abstract

A well-separated pair decomposition (WSPD) of a set $P$ of $n$ points covers all $\binom{n}{2}$ pairs of points by a family of dumbbells $\{X,Y\}$, each covering the pairs of $X\times Y$, such that the diameters of $X$ and $Y$ are at most an $\varepsilon$-fraction of their distance. In $\mathbb{R}^d$, WSPDs with $O(n/\varepsilon^d)$ dumbbells always exist, but many inputs admit much smaller ones. However, computing a WSPD of minimum size is NP-hard in the plane. For $d\ge2$, the only known approximation that preserves the separation has a factor of $2^{O(d)}$, and Buchin et al. (ESA 2026) asked whether a $(1+δ)$-approximation exists. We give a bicriteria approximation scheme: for fixed $d$ and $δ\in(0,1]$, we compute in $f(d,δ)\cdot n^{6}$ time a $(1+δ)\varepsilon$-WSPD whose size is at most $(1+δ)$ times the minimum size of an $\varepsilon$-WSPD. The same holds in metric spaces of bounded doubling dimension. The algorithm combines shifting over scales with the local-growth technique of Nieberg, Hurink, and Kern, applied to a graph whose vertices are the pairs of points. In contrast, we show that in general metric spaces, relaxing the separation by a factor below $2$ keeps the problem hard: for every fixed rational $\varepsilon\le1/4$, it is NP-hard to compute such a relaxed WSPD whose size is within a factor $n^c$ of the minimum, for every $c<2/3$.