We propose a randomized approximation scheme for the Euclidean Steiner Multi Cycle problem which runs in quasilinear time. In this problem, we are given a set of n pairs of points (terminals) $\mathcal{T}={\{\{t_i,t_i^{\prime }\}\}}_{i=1}^n$ in the Euclidean plane, and the objective is to find a collection of cycles of minimum cost such that $t_i$ and $t_i^{\prime }$ belong to a same cycle, for each $i\in \{1,\dots ,n\}$. This problem extends the Steiner Cycle problem in the same way the Steiner Forest extends the Steiner Tree problem. Additionally, it has applications on routing problems with pickup and delivery locations.

中文翻译：

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欧氏平面上Steiner多周期的随机近似方案

针对拟线性时间的欧氏Steiner多周期问题，我们提出了一个随机近似方案。在这个问题上，我们得到了一组n对点（端点）$\mathcal{Ť}={\{\{{Ť}_{\mathrm{一世}}，{Ť}_{\mathrm{一世}}^{\prime }\}\}}_{\mathrm{一世}=\mathrm{1个}}^{ñ}$ 在欧几里得平面上，目标是找到最小成本的周期集合，使得 ${Ť}_{\mathrm{一世}}$ 和 ${Ť}_{\mathrm{一世}}^{\prime }$ 属于同一周期，对于每个 $\mathrm{一世}\in \{\mathrm{1个}，\dots ，ñ\}$。该问题以Steiner Forest扩展Steiner Tree问题的相同方式扩展了Steiner Cycle问题。此外，它还具有有关取件和交付位置的路由问题的应用程序。