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arXiv · 2507.19649

Online Rounding Schemes for $ k $-Rental Problems

Hossein Nekouyan·Bo Sun·Raouf Boutaba·Xiaoqi Tan
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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 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 kRental-Variable, 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.

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Hossein Nekouyan, Bo Sun, Raouf Boutaba, Xiaoqi Tan. 2025-07-25. Online Rounding Schemes for $ k $-Rental Problems. https://arxiv.org/abs/2507.19649

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