Bounds for k-centers of point sets under $L_{\infty}$-bottleneck distance

TL;DR

The paper studies k-center clustering for fixed-size point sets in the plane under the $L_\infty$-bottleneck distance, motivated by persistence diagrams and reframed as the Restaurant Supply Problem. Under the key assumption that any two restaurant chains can be $\delta$-satisfied by a single supermarket chain, it derives both an upper bound and a lower bound on the required number of supermarket chains, highlighting a sharp exponential dependence on the chain size $m$ and ruling out subexponential bounds. It also analyzes computational complexity, providing a polynomial-time algorithm for the two-store-per-chain case and establishing NP-completeness for unbounded $m$ through a planar 3-SAT reduction, while noting the problem remains tractable for fixed $m$ but becomes intractable as $m$ grows. Overall, the work connects geometric covering, Helly-type results for axis-aligned boxes, and complexity theory in a novel setting inspired by persistence diagrams and supply-chain analogies.

Abstract

We consider the $k$-center problem on the space of fixed-size point sets in the plane under the $L_{\infty}$-bottleneck distance. While this problem is motivated by persistence diagrams in topological data analysis, we illustrate it as a \emph{Restaurant Supply Problem}: given $n$ restaurant chains of $m$ stores each, we want to place supermarket chains, also of $m$ stores each, such that each restaurant chain can select one supermarket chain to supply all its stores, ensuring that each store is matched to a nearby supermarket. How many supermarket chains are required to supply all restaurants? We address this questions under the constraint that any two restaurant chains are close enough under the $L_{\infty}$-distance to be satisfied by a single supermarket chain. We provide both upper and lower bounds for this problem and investigate its computational complexity.

Figures (6)

• Figure 1: Two restaurants annotated with the same index from chains in the same permutation split the plane into quadrants with the same number of restaurants.
• Figure 2: A city with 3 restaurant chains, each consisting of 2 stores placed on antipodal points.
• Figure 3: Placing supermarkets in the green (red) regions sets the variable to "true" ("false").
• Figure 4: A clause gadget.
• Figure 5: All gadgets for the (very small) formula $\varphi=(x_1\vee\neg x_2\vee x_3)\wedge(\neg x_1\vee x_2\vee x_4)$. The black points describe a placement of supermarkets corresponding to setting all variables to "true".

Theorems & Definitions (13)

• Definition 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm:upper_bound']}
• Theorem 3.1
• proof
• Lemma 4.1
• proof
• Theorem 4.2