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

2025-03-04

AI Summary

研究在平面上固定大小点集的$k$-中心问题，使用$L_{infty}$瓶颈距离，提供上界和下界，并探讨计算复杂性。

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Bounds for k-centers under L∞-bottleneck distance
• Restaurant Supply Problem with proximity constraints
• Computational complexity of supermarket chain placement

Innovation

• Uses L∞-bottleneck distance for k-center problem
• Models as Restaurant Supply Problem with constraints
• Provides bounds and computational complexity analysis