Online Rounding Schemes for $ k $-Rental Problems

Abstract: We study two online resource-allocation problems with reusability in an adversarial setting, namely kRental-Fixed and kRental-Variable. 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 kRental-Fixed, we present an optimal randomized algorithm that attains the best possible competitive ratio: it first computes an optimal fractional allocation via a price-based approach, then applies a novel lossless online rounding scheme to obtain an integral solution. For kRental-Variable, we prove that lossless online rounding is impossible. We introduce a limited-correlation rounding technique that treats each unit independently while introducing controlled dependencies across allocation decisions involving the same unit. Coupled with a carefully crafted price-based method for computing the fractional allocation, this yields an order-optimal competitive ratio for the variable-duration setting.