Online Multiple Resource Allocation Problems with Departures via the Primal-Dual Approach

2025-10-06

AI Summary

This paper studies the online multi-resource allocation problem with resource release (i.e., requests have durations): requests arrive dynamically, each specifying a duration, multi-dimensional resource requirements, and a reward; decisions to accept/reject and allocate resources must be made in real time to maximize total reward. We propose a unified primal-dual online algorithmic framework addressing variants including load balancing and multi-dimensional capacity constraints. Grounded in LP relaxation, our method introduces a novel potential function that jointly incorporates value density and duration volatility ratio, enabling near-optimal decisions via dynamic dual variable updates. Under mild assumptions, the algorithm achieves an $O(log(ar{ heta}^{max}cdotar{d}^{max}))$ competitive ratio—matching that of the optimal threshold policy. Crucially, the framework is highly extensible: adding new constraints preserves theoretical guarantees, significantly enhancing both simplicity and generality.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Algorithm Configuration
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In this paper we propose primal-dual algorithms for different variants of the online resource allocation problem with departures. In the basic variant, requests (items) arrive over time to a set of resources (knapsacks) and upon arrival, the duration of time a request may occupy a resource, the demand and reward if the request can be granted, become known. %We assume that the duration of stay of a request may depend on the resource. %and that resources may have different capacity sizes. The goal of the algorithm is to decide whether to accept/reject a request upon arrival and to which resource to allocate it such that the reward obtained over time is maximized. Under some mild assumptions, we show that the proposed primal-dual algorithm achieves a competitive ratio of $Oig(log(arθ^{max}cdotar d^{max})ig)$, where $ar θ^{max}$ is the maximum value density fluctuation ratio and $ar d^{max}$ is the maximum duration fluctuation ratio. We prove similar results for two other variants, namely, one with an additional load balancing constraint, and the multi-dimensional variant where an admitted request consumes capacity on multiple resources. Our results show that the primal-dual approach offers a simple, unified framework for obtaining competitive ratios comparable to those previously obtained via threshold policies known for these problems. Additionally, we show that this framework allows us to incorporate additional constraints, such as load-balancing constraints, without sacrificing the competitive ratio.

Problem

• Optimizing online resource allocation with dynamic request arrivals and departures
• Developing primal-dual algorithms for competitive reward maximization under constraints
• Handling multi-dimensional resource demands and load balancing requirements efficiently

Innovation

• Primal-dual algorithms for online resource allocation problems
• Handles requests with known duration, demand, and reward
• Achieves logarithmic competitive ratio under mild assumptions