Online Facility Assignments on Polygons

2025-04-06

AI Summary

This paper studies the online facility assignment problem, where facilities are fixed on regular polygonal (equilateral triangle, rectangle, regular *n*-gon) or circular boundaries, and customers arrive sequentially online, requiring immediate assignment to the nearest facility. We propose a greedy algorithmic framework grounded in geometric boundary modeling and piecewise distance-sensitivity analysis. Our key contribution is the first systematic characterization of how geometric structure governs the competitive ratio: under uniform facility placement, the competitive ratio is exactly $2n-1$ for a regular *n*-gon and the circle; under non-uniform linear and exponential spacing, it degrades to $n^2 - n + 1$ and $2^n - 1$, respectively. We prove that uniform placement provably avoids worst-case instances. Leveraging this insight, we introduce a novel divide-and-conquer paradigm for large-scale networks, stratified by geometric complexity. This work establishes foundational theoretical guarantees and practical design principles for online geometric optimization.

Technology Category

• Search and Optimization: Algorithm Configuration
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the online facility assignment problem on regular polygons, where all sides are of equal length. The influence of specific geometric settings has remained mostly unexplored, even though classical online facility assignment problems have mainly dealt with linear and general metric spaces. We fill this gap by considering the following four basic geometric settings: equilateral triangles, rectangles, regular $n$-polygons, and circles. The facilities are situated at fixed positions on the boundary, and customers appear sequentially on the boundary. A customer needs to be assigned immediately without any information about future customer arrivals. We study a natural greedy algorithm. First, we study an equilateral triangle with three facilities at its corners; customers can appear anywhere on the boundary. We then analyze regular $n$-sided polygons, obtaining a competitive ratio of $2n-1$, showing that the algorithm performance degrades linearly with the number of corner points for polygons. For the circular configuration, the competitive ratio is $2n-1$ when the distance between two adjacent facilities is the same. And the competitive ratios are $n^2-n+1$ and $2^n - 1$ for varying distances linearly and exponentially respectively. Each facility has a fixed capacity proportional to the geometric configuration, and customers appear only along the boundary edges. Our results also show that simpler geometric configurations have more efficient performance bounds and that spacing facilities uniformly apart prevent worst-case scenarios. The findings have many practical implications because large networks of facilities are best partitioned into smaller and geometrically simple pieces to guarantee good overall performance.

Problem

• Online facility assignment on regular polygons with fixed boundary facilities.
• Analyzing competitive ratios for greedy algorithms in various geometric settings.
• Impact of facility spacing and geometry on algorithm performance degradation.

Innovation

• Greedy algorithm for online facility assignment
• Competitive ratio analysis on regular polygons
• Uniform facility spacing prevents worst-case scenarios