On the hardness of computing an average curve.

We study the complexity of clustering curves under $k$-median and $k$-center objectives in the metric space of the Fru0027echet distance and related distance measures. The $k$-center problem has recently been shown to be NP-hard, even in the case where $k=1$, i.e. the minimum enclosing ball under the Fru0027echet distance. We extend these results by showing that also the $k$-median problem is NP-hard for $k=1$. 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 Fru0027echet and Dynamic Time Warping (DTW) distance. Our result generalizes an earlier result by Bulteau et al. from 2018 for a variant of DTW that uses squared distances. Moreover, closing some gaps in the literature, we show positive results for a variant where the center curve may have complexity at most $ell$ under the discrete Fru0027echet 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.