Fine-Grained Complexity of Continuous Euclidean k-Center

Abstract

In the (continuous) Euclidean $k$-center problem, given $n$ points in $\mathbb{R}^d$ and an integer $k$, the goal is to find $k$ center points in $\mathbb{R}^d$ that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. $\bullet$ Parameterized by $k$: Assuming the Exponential Time Hypothesis (ETH), we show that there is no $f(k)n^{o(k^{1-1/d})}$-time algorithm for the Euclidean $k$-center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any $(1+\varepsilon)$-approximation algorithm running in time $(k/\varepsilon)^{o(k^{1-1/d})}n^{O(1)}$, thereby establishing near-optimality of the corresponding approximation scheme by the same authors. $\bullet$ Small $k$: Assuming the 3-SUM hypothesis, we prove that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the Euclidean $2$-center problem in $\mathbb{R}^3$. This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any $\varepsilon > 0$, the Euclidean $6$-center problem in $\mathbb{R}^2$ also admits no $O(n^{2-\varepsilon})$-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

Figures (16)

• Figure 1: (a) Three sets of points $\widetilde{{\mathcal{A}}},\widetilde{{\mathcal{B}}},\widetilde{{\mathcal{C}}}$ constructed from $X$. The point sets $\widetilde{{\mathcal{A}}},\widetilde{{\mathcal{B}}}, \widetilde{{\mathcal{C}}}$ are oriented upward. (b) Two approximate unit balls associated with a solution of the Gap Convolution 3SUM problem, encoding a 2-clustering for the point sets $\widetilde{{\mathcal{A}}}$, $\widetilde{{\mathcal{B}}}$, $\widetilde{{\mathcal{C}}}$, and anchor points $d_i^+$ and $d_i^{-}$ for $i \in \{0,\ldots,4\}$.
• Figure 2: (left) The point sets $\widetilde{{\mathcal{A}}}_1$, $\widetilde{{\mathcal{B}}}_1$, and $\widetilde{{\mathcal{C}}}$ are defined on edges that contain the point $(0,1)$. (right) The upward movement of $D_1$ and $D_4$ induces the movements of other disks. Anchor points are not shown.
• Figure 3: The basic curve is a translation of the curve $S$ on the left. On the right, points are placed along the basic curve $S_u$ of some vertex $u$, and disks that cover these points. The orientation is counterclockwise.
• Figure 4: (a) A grid graph in $\mathbb{R}^2$ with $k = 9$ vertices; (b) A schematic view of the reduction; the shared edges marked in (a) are highlighted. Each arrow denotes the orientation of the edges of the basic curve. (c) Copies of nonoverlapping basic curves with the same incidence structure as the full grid graph. Translational symmetries are indicated by the green arrows.
• Figure 5: The hexagonal (center) vertices are drawn in purple and the anchor points are blue. The vertex $h$ is on the boundary of the convex hull of $\mathcal{H}$. Two disks of radius $2(1-\varepsilon+\varepsilon^{1.7}\xspace)$, centered at points of the convex hull of the hexagonal vertices, are shown in orange. The shaded points are those that can be covered by either disk.

Theorems & Definitions (54)

• Theorem 1: Exact $k\text{-}\mathsf{center}$ in $\mathbb{R}^d$
• Theorem 2: Approximate $k\text{-}\mathsf{center}$ in $\mathbb{R}^d$
• Theorem 3: 2-center problem in $\mathbb{R}^3$
• Theorem 4: 6-center problem in $\mathbb{R}^2$
• Theorem 5
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3