在线合同选择以实现持续覆盖
Online Contract Selection for Continual Coverage
AI总结：
研究在线合同选择问题，针对不确定价格下的持续覆盖需求，提出两种模型（延迟模型和并发模型），通过量化方法设计算法，改进了竞争比率的上下界。
摘要：
受系统需通过持续采购合同保持运行的应用启发，我们研究了两个在线合同选择问题，在不确定价格下。在每个时间步，从已知分布中揭示的价格被在线显示，决策者可发起任意持续时间的合同，其成本等于价格与合同长度的乘积；且每个时间周期必须由至少一个活跃合同覆盖。我们考虑两种模型，取决于合同如何覆盖时间：一种是延迟模型，合同按顺序排队；另一种是并发模型，合同立即生效且可能重叠。在两种设置中，我们寻求最小化竞争比率的在线算法，即在线算法所产生成本与已知所有价格时的最优成本的比率。我们首先关注独立同分布的价格情况。对于延迟模型，我们精确地刻画了最坏情况下的最优竞争比率，当时间跨度增长时，其渐近值约为ζ*≈2.472。对于并发模型，我们证明了最优竞争比率的下界为ζ*，并证明了渐近竞争比至多为4.179。这些界值改进了当前最优竞争比率的下界2.148和上界6.052。对于两种模型，我们的算法基于分位数，可以轻松转换为任何分布的实用阈值算法。我们的证明基于分位数空间中的线性规划和对偶论点。最后，我们证明，当价格仍独立但不一定同分布时，在两种模型中不存在有限的竞争比率，证明了两种价格设置之间的显著区别。
英文摘要：
Motivated by applications where a system must remain operational via continual procurement of contracts, we study two online contract selection problems under uncertain prices. At each time step, a price drawn from a known distribution is revealed online, and the decision-maker may initiate a contract of arbitrary duration, incurring a cost equal to the product of the price and the contract length; moreover, every time period must be covered by at least one active contract. We consider two models depending on how contracts cover time: a \emph{deferred model}, in which contracts are queued back-to-back, and a \emph{concurrent model}, in which contracts become active immediately and may overlap. In both settings, we seek online algorithms that minimize their competitive ratio, i.e., the ratio between the expected cost incurred by the online algorithm and the expected offline optimal cost when all prices are known in advance. We first focus on the case where prices are independent and identically distributed (i.i.d.). For the deferred model, we characterize exactly the worst-case optimal competitive ratio, which is asymptotically $ζ^* \approx 2.472$ as the time horizon grows. For the concurrent model, we prove a lower bound of $ζ^*$ on the optimal competitive ratio and an asymptotic competitive ratio of at most $4.179$. These bounds improve upon the current lower bound of $2.148$ and upper bound of $6.052$ on the optimal competitive ratio. For both models, our algorithms are quantile-based that can be easily translated into practical threshold-based algorithms for any distribution. Our proofs follow from linear programs and duality arguments in quantile spaces. Lastly, we show that, in both models, no finite competitive ratio exists when the prices are still independent but not necessarily identically distributed, proving a striking division in the two price settings.
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