$\newcommand{\Re}{\mathbb{R}}$We study the minWSPD problem of computing the minimum-size well-separated pairs decomposition of a set of points, and show constant approximation algorithms in low-dimensional Euclidean space and doubling metrics. This problem is computationally hard already $\Re^2$, and is also hard to approximate. We also introduce a new pair decomposition, removing the requirement that the diameters of the parts should be small. Surprisingly, we show that in a general metric space, one can compute such a decomposition of size $O( \tfrac{n}{\varepsilon}\log n)$, which is dramatically smaller than the quadratic bound for WSPDs. In $\Re^d$, the bound improves to $O( d \tfrac{n}{\varepsilon}\log \tfrac{1}{\varepsilon } )$.

翻译：$\newcommand{\Re}{\mathbb{R}}$我们研究了计算点集最小规模良好分离对分解的minWSPD问题，并给出了低维欧几里得空间及加倍度量空间中该问题的常数近似算法。该问题在$\Re^2$中已是计算困难的，且难以近似。我们还提出了一种新的对分解方法，取消了各部分直径必须很小的限制。令人惊讶的是，我们证明在一般度量空间中，可以计算出规模为$O( \tfrac{n}{\varepsilon}\log n)$的此类分解，这显著小于WSPD的二次界。在$\Re^d$中，该界可改进为$O( d \tfrac{n}{\varepsilon}\log \tfrac{1}{\varepsilon } )$。