超越均匀开设成本的随机顺序在线设施选址
Random-Order Online Facility Location Beyond Uniform Opening Costs
AI总结：
研究随机顺序模型下任意正开设成本的在线度量设施选址，给出确定性4.2674竞争算法，改进此前结果，还证明3 - o(1)随机算法下界，揭示全空间均匀与非均匀成本模型的严格分离。
AI中文摘要：
我们研究具有任意正开设成本的随机顺序模型中的在线度量设施选址问题。预先已知一组有限的候选设施及其成本，对手确定一个以均匀随机顺序到达的需求点多重集。此设置包括规定的候选站点和经典的有限全空间节点成本模型。对于已知的时间范围，我们给出一种确定性的4.2674竞争算法，改进了之前非均匀开设成本下的33倍因子。在秩t时，算法使用正归一化秩qt = t / n，选择使d(x,y)+λtfy最小的候选设施，其中λt = min{1,qt / μ}，并在当前连接距离覆盖此惩罚目标时开设它。分析使用单调单轮收费和上包络分解来控制后续点和每个最优聚类的第一个点。对于单位开设成本，该规则精确地简化为对最近候选设施可实现的距离改进的截止值。补充附录给出了对密切相关的零起始秩截止的更精确分析，并获得了低于3.2805的比率。我们还证明了任意随机在线算法的3 - o(1)下界。该下界在规定候选集上具有均匀成本时已经成立，并且无损地转移到具有非均匀开设成本的有限全空间模型。连同最近全空间均匀成本下低于2.42的竞争比率，这在全空间均匀成本和非均匀成本模型之间产生了严格的分离。
英文摘要：
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.
