Near-Optimal Dimension Reduction for Facility Location

Oblivious dimension reduction, \`{a} la the Johnson-Lindenstrauss (JL) Lemma, is a fundamental approach for processing high-dimensional data. We study this approach for Uniform Facility Location (UFL) on a Euclidean input $X\subset\mathbb{R}^d$, where facilities can lie in the ambient space (not restricted to $X$). Our main result is that target dimension $m=\tilde{O}(\epsilon^{-2}\mathrm{ddim})$ suffices to $(1+\epsilon)$-approximate the optimal value of UFL on inputs whose doubling dimension is bounded by $\mathrm{ddim}$. It significantly improves over previous results, that could only achieve $O(1)$-approximation [Narayanan, Silwal, Indyk, and Zamir, ICML 2021] or dimension $m=O(\epsilon^{-2}\log n)$ for $n=|X|$, which follows from [Makarychev, Makarychev, and Razenshteyn, STOC 2019]. Our oblivious dimension reduction has immediate implications to streaming and offline algorithms, by employing known algorithms for low dimension. In dynamic geometric streams, it implies a $(1+\epsilon)$-approximation algorithm that uses $O(\epsilon^{-1}\log n)^{\tilde{O}(\mathrm{ddim}/\epsilon^{2})}$ bits of space, which is the first streaming algorithm for UFL to utilize the doubling dimension. In the offline setting, it implies a $(1+\epsilon)$-approximation algorithm, which we further refine to run in time $( (1/\epsilon)^{\tilde{O}(\mathrm{ddim})} d + 2^{(1/\epsilon)^{\tilde{O}(\mathrm{ddim})}}) \cdot \tilde{O}(n) $. Prior work has a similar running time but requires some restriction on the facilities [Cohen-Addad, Feldmann and Saulpic, JACM 2021]. Our main technical contribution is a fast procedure to decompose an input $X$ into several $k$-median instances for small $k$. This decomposition is inspired by, but has several significant differences from [Czumaj, Lammersen, Monemizadeh and Sohler, SODA 2013], and is key to both our dimension reduction and our PTAS.