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

TL;DR

This paper develops a unified primal-dual framework for online resource allocation problems with departures, including the basic OMKD, a load-balancing variant, and a multidimensional variant. By formulating each problem as an IP with an LP relaxation and its dual, the authors design online algorithms that update dual prices exponentially and select allocations to maintain near-optimal dual primal progress. They prove competitive ratios of $O\left(\log(\bar{\theta}^{\max}\cdot\bar{d}^{\max})\right)$ for the basic and lbOMKD variants and $O\left(\log(\bar{\rho}^{\max}\cdot\xi^{\max}\cdot\bar{d}^{\max})\right)$ for MdOMKD, matching or improving prior threshold-based results under comparable assumptions. The framework also naturally accommodates additional constraints like load balancing without worsening the CR, offering a simple, scalable approach to complex online packing problems with departures and multiple resource types.

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 $O\big(\log(\barθ^{\max}\cdot\bar d^{\max})\big)$, where $\bar θ^{\max}$ is the maximum value density fluctuation ratio and $\bar d^{\max}$ 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.

Figures (1)

• Figure 1: Illustration of the worst-case of final capacity utilization.

Theorems & Definitions (12)

• Theorem 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Theorem 5.1
• Lemma 5.2
• Lemma 5.3
• Lemma 5.4
• Theorem 6.1
• Lemma 6.2