Online Rounding Schemes for $ k $-Rental Problems
Hossein Nekouyan, Bo Sun, Raouf Boutaba, Xiaoqi Tan

TL;DR
This paper introduces new online algorithms with provable guarantees for resource allocation problems involving reusable units, achieving optimal or near-optimal competitive ratios in adversarial settings.
Contribution
It develops the first optimal randomized algorithm for kRental-Fixed and a novel limited-correlation rounding technique for kRental-Variable, advancing online resource allocation methods.
Findings
Optimal randomized algorithm for kRental-Fixed with best competitive ratio.
Limited-correlation rounding technique for kRental-Variable with order-optimal performance.
Theoretical guarantees established for both fixed and variable duration settings.
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 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…
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