Near-Optimal Dimension Reduction for Facility Location

TL;DR

This paper presents a near-optimal oblivious dimension reduction technique for Uniform Facility Location problems in Euclidean spaces with bounded doubling dimension, improving approximation ratios and computational efficiency for streaming and offline algorithms.

Contribution

It introduces a dimension reduction method that achieves a (1+ε)-approximation for UFL with dimension m=~O(ε^{-2} ddim), surpassing previous bounds and enabling efficient streaming and offline algorithms.

Findings

Achieves (1+ε)-approximation with dimension m=~O(ε^{-2} ddim)

First streaming algorithm for UFL utilizing doubling dimension

Refined algorithm with improved runtime for offline UFL

Abstract

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.