Online Rounding and Pricing Schemes for k-Rental Problems

摘要

We study two online resource allocation problems with reusability in an adversarial setting, namely \problemkRentalFD and \problemkRentalVD. In both problems, a decision-maker manages k identical reusable units and faces a sequence of rental requests over time. We develop theoretically grounded relax-and-round algorithms with provable competitive ratio guarantees for both settings. For \problemkRentalFD, we present an optimal randomized algorithm that achieves the best possible competitive ratio. The algorithm first computes an optimal fractional allocation using a price-based approach, and then applies a novel lossless online rounding scheme to obtain an integral solution. For \problemkRentalVD, we first establish the impossibility of achieving lossless online rounding. We then introduce a limited-correlation rounding technique that treats each unit independently while introducing controlled dependencies across allocation decisions involving the same unit. Combined with a carefully-crafted price-based method for computing the fractional allocation, this approach yields an order-optimal competitive ratio for the variable-duration setting.