Online Rounding Schemes for $ k $-Rental Problems

2025-07-25

AI Summary

This paper studies online resource allocation under adversarial conditions with reusable resources, focusing on two leasing models: kRental-Fixed (fixed lease duration) and kRental-Variable (variable lease duration), where a decision-maker must allocate k identical reusable units to dynamically arriving lease requests. We propose a price-based fractional allocation framework, augmented by novel lossless online rounding and bounded-correlation rounding techniques—introducing controlled dependence among unit assignments while preserving per-unit independent processing, thereby overcoming performance bottlenecks of traditional independent rounding. Theoretically, we achieve the optimal randomized competitive ratio for kRental-Fixed and an asymptotically optimal competitive ratio for kRental-Variable. To our knowledge, this is the first work to establish tight theoretical bounds for reusable leasing problems, significantly enhancing both the efficiency and robustness of online decision-making.

Technology Category

• Machine Learning: Online Learning & Bandits
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Develop online algorithms for k reusable resource allocation
• Achieve optimal competitive ratio for fixed-duration rentals
• Address impossibility of lossless rounding in variable-duration rentals

Innovation

• Optimal randomized algorithm with price-based approach
• Novel lossless online rounding scheme
• Limited-correlation rounding technique with controlled dependencies