Online Contract Selection for Continual Coverage

2026-05-15

AI Summary

This study addresses the online contract duration selection problem under price uncertainty, aiming to continuously cover all time points while minimizing total cost. Focusing on both the delayed and concurrent models, the authors propose quantile-based threshold strategies and employ linear programming together with duality analysis. Under the assumption of independent and identically distributed (i.i.d.) prices, they precisely characterize the asymptotically optimal competitive ratio for the delayed model (approximately 2.472) and significantly improve the upper bound for the concurrent model from 6.052 to 4.179. Furthermore, they establish that no algorithm can achieve a finite competitive ratio when prices are non-i.i.d., thereby delineating a fundamental theoretical boundary for feasibility in this setting.

Abstract

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.

Problem

• online contract selection
• continual coverage
• competitive ratio
• uncertain prices
• stochastic optimization

Innovation

• online contract selection
• competitive ratio
• quantile-based algorithm
• stochastic optimization
• continual coverage