Random-Order Online Facility Location Beyond Uniform Opening Costs

2026-07-24

AI Summary

This study addresses the online metric facility location problem with non-uniform opening costs under stochastic demand arrival order. Assuming known candidate facilities and arbitrary positive opening costs, the authors propose a deterministic online algorithm that employs a normalization-based ranking scheme to define a dynamic penalty objective, combined with monotone single-round cost allocation and upper-envelope decomposition techniques. The algorithm significantly improves the competitive ratio for non-uniform costs from 33 to 4.2674, and further reduces it below 3.2805 in the uniform-cost setting. Additionally, the paper establishes a lower bound of $3 - o(1)$ on the competitive ratio for any randomized algorithm, thereby demonstrating a strict performance gap between the uniform and non-uniform cost models.

Abstract

We study online metric facility location in the random-order model with arbitrary positive opening costs. A finite set of candidate facilities and their costs is known in advance, while an adversary fixes a multiset of demand points that arrives in a uniformly random order. This setting includes both prescribed candidate sites and the classical finite full-space node-cost model. For a known horizon, we give a deterministic $4.2674$-competitive algorithm, improving the previous factor $33$ for nonuniform opening costs. At rank $t$, the algorithm uses the positive normalized rank $q_t=t/n$, chooses a candidate minimizing $d(x,y)+λ_t f_y$, where $λ_t=\min\{1,q_t/μ\}$, and opens it when the current connection distance covers this penalized objective. The analysis uses a monotone one-round charge and an upper-envelope decomposition to control later points and the first point of each optimal cluster. With unit opening costs, the rule reduces exactly to a cutoff on the distance improvement attainable from a nearest candidate. A supplementary appendix gives the sharper analysis of the closely related zero-start rank cutoff and obtains a ratio below $3.2805$. We also prove a $3-o(1)$ lower bound for arbitrary randomized online algorithms. The lower bound already holds with uniform costs on a prescribed candidate set and transfers, without loss, to the finite full-space model with nonuniform opening costs. Together with the recent competitive ratio below $2.42$ for full-space uniform costs, this yields a strict separation between the full-space uniform- and nonuniform-cost models.

Problem

• online facility location
• random-order model
• nonuniform opening costs
• competitive analysis
• metric space

Innovation

• online facility location
• random-order model
• nonuniform opening costs
• competitive analysis
• deterministic algorithm