Approximating (k,l)-center clustering for curves

The Euclidean k-Center problem is a classical problem that has been extensively studied in computer science. Given a set G of n points in Euclidean space, the problem is to determine a set C of k centers (not necessarily part of G) such that the maximum distance between a point in G and its nearest neighbor in C is minimized. In this paper we study the corresponding (k, l)-CENTER problem for polygonal curves under the Frechet distance, that is, given a set G of n polygonal curves in ℝd, each of complexity m, determine a set C of k polygonal curves in ℝd, each of complexity l, such that the maximum Frechet distance of a curve in G to its closest curve in C is minimized. In their 2016 paper, Driemel, Krivosija, and Sohler give a near-linear time (1 + e-approximation algorithm for one-dimensional curves, assuming that k and l are constants. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension 2 and higher. Our analysis thus extends to application-relevant input data such as GPS-trajectories and protein backbones. We show that, if l is part of the input, then there is no polynomial-time approximation scheme unless P = NP. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Frechet distance. In the case of the discrete Frechet distance on two-dimensional curves, we show hardness of approximation within a factor close to 2.598. This result also holds when k = 1, and the NP-hardness extends to the case that l = ∞, i.e., for the problem of computing the minimum-enclosing ball under the Frechet distance. Finally, we observe that a careful adaptation of Gonzalez’ algorithm in combination with a curve simplification yields a 3-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight. Read More: https://epubs.siam.org/doi/10.1137/1.9781611975482.181