Online Rounding and Pricing Schemes for k-Rental Problems

We study two online resource allocation problems with reusability in an adversarial setting, namely and . 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 , 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 , 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.