关于平均曲线计算的困难性
On the hardness of computing an average curve
摘要：
我们研究了在Fréchet距离和相关距离测度的度量空间中$k$ -median和$k$ -center目标下聚类曲线的复杂性。最近的硬度的最小封闭球问题的Fréchet距离下的结果的基础上，我们表明，也$1$ -中位数问题是NP-困难的。进一步证明了以曲线个数为参数的$1$ -中值问题是W[1]-困难的.我们在离散和连续的Fréchet和动态时间弯曲（DTW）距离下证明了这一点。这产生了布尔托等人2018年早期结果的独立证明，用于使用平方距离的DTW变体，其中新的证明更简单，更通用。在积极的一面，我们给近似算法的问题的变种，中心曲线可能有复杂性最多$\ell$下的离散Fréchet距离。特别是，对于固定的$k,\ell$和$\varepsilon$，我们给$(1+\varepsilon)$ -近似算法的$(k,\ell)$ -中位数和$(k,\ell)$ -中心的目标和多项式时间精确算法的$(k,\ell)$ -中心的目标。
英文摘要：
We study the complexity of clustering curves under $k$-median and $k$-center objectives in the metric space of the Fréchet distance and related distance measures. Building upon recent hardness results for the minimum-enclosing-ball problem under the Fréchet distance, we show that also the $1$-median problem is NP-hard. Furthermore, we show that the $1$-median problem is W[1]-hard with the number of curves as parameter. We show this under the discrete and continuous Fréchet and Dynamic Time Warping (DTW) distance. This yields an independent proof of an earlier result by Bulteau et al. from 2018 for a variant of DTW that uses squared distances, where the new proof is both simpler and more general. On the positive side, we give approximation algorithms for problem variants where the center curve may have complexity at most $\ell$ under the discrete Fréchet distance. In particular, for fixed $k,\ell$ and $\varepsilon$, we give $(1+\varepsilon)$-approximation algorithms for the $(k,\ell)$-median and $(k,\ell)$-center objectives and a polynomial-time exact algorithm for the $(k,\ell)$-center objective.
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