Online Rounding Schemes for $ k $-Rental Problems

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 $k$ 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.