[Paper Review] Online Multidimensional Packing Problems in the Random-Order Model

David Naori, Danny Raz|arXiv (Cornell University)|Jul 1, 2019

Optimization and Search Problems19 references4 citations

TL;DR

This paper presents an online algorithm for the $d$-dimensional Vector Generalized Assignment Problem (VGAP) in the random-order model, using a three-phase approach combining sampling, maximum-weight bipartite matching for heavy items, and LP relaxation with randomized rounding for light items. It achieves an $O(d)$-competitive ratio, which is asymptotically optimal, and improves upon prior competitive ratios for both one-dimensional GAP and knapsack problems.

ABSTRACT

We study online multidimensional variants of the generalized assignment problem which are used to model prominent real-world applications, such as the assignment of virtual machines with multiple resource requirements to physical infrastructure in cloud computing. These problems can be seen as an extension of the well known secretary problem and thus the standard online worst-case model cannot provide any performance guarantee. The prevailing model in this case is the random-order model, which provides a useful realistic and robust alternative. Using this model, we study the $d$-dimensional generalized assignment problem, where we introduce a novel technique that achieves an $O(d)$-competitive algorithms and prove a matching lower bound of $Ω(d)$. Furthermore, our algorithm improves upon the best-known competitive-ratio for the online (one-dimensional) generalized assignment problem and the online knapsack problem.

Motivation & Objective

• To address online multidimensional packing problems in cloud computing and network function virtualization, where traditional worst-case models fail due to poor competitive ratios.
• To design a robust online algorithm for the $d$-dimensional Vector Generalized Assignment Problem (VGAP) under the random-order model, which better reflects real-world arrival patterns.
• To overcome the limitations of prior algorithms that exclude either heavy or light items by handling both types simultaneously in a unified framework.
• To establish tight theoretical bounds by proving an $\Omega(d)$ lower bound, matching the upper bound of $O(d)$, demonstrating asymptotic optimality.
• To extend the framework to special cases like $\{0,1\}$-VGAP and Vector Multiple Knapsack Problem (VMKP), showing broad applicability.

Proposed method

• The algorithm operates in three phases: a sampling phase to observe initial items without packing, followed by dedicated phases for heavy and light packing options.
• For heavy items (those consuming more than half a bin's capacity in any dimension), the algorithm uses maximum-weight bipartite matching to compute tentative assignments.
• For light items (those consuming at most half a bin's capacity), the algorithm computes an optimal fractional solution to the LP relaxation of the local packing problem and applies randomized rounding.
• The algorithm ensures feasibility by only packing an item if its tentative assignment fits in the bin’s remaining capacity, based on the local solution.
• The framework is extended to $\{0,1\}$-VGAP by partitioning based on the number of non-zero entries in the consumption vector, and to VMKP via bin sorting by capacity.
• Theoretical analysis uses probabilistic arguments and matrix constructions to derive lower bounds, leveraging independence and capacity constraints across dimensions.

Experimental results

Research questions

• RQ1 Can an online algorithm achieve a competitive ratio independent of the number of items for $d$-dimensional generalized assignment in the random-order model?
• RQ2 Is it possible to design an algorithm that packs both heavy and light items effectively, avoiding the exclusion of one class as in prior work?
• RQ3 What is the optimal competitive ratio achievable for online multidimensional packing problems, and can it be matched by a lower bound?
• RQ4 How does the competitive ratio scale with dimensionality $d$, and is $O(d)$ the tightest possible bound?
• RQ5 Can the framework be adapted to special cases like $\{0,1\}$-VGAP and VMKP while preserving optimality?

Key findings

• The proposed algorithm achieves an $O(d)$-competitive ratio for the $d$-dimensional Vector Generalized Assignment Problem (VGAP) in the random-order model.
• A matching lower bound of $\Omega(d)$ is proven for the vector knapsack problem, showing that the $O(d)$ competitive ratio is asymptotically optimal.
• The algorithm improves upon the best-known competitive ratio for the one-dimensional online generalized assignment problem and the online knapsack problem.
• For the special case of $\{0,1\}$-VGAP, the algorithm uses a partitioning criterion based on the number of non-zero entries in the consumption vector to manage packing decisions.
• The framework is extendable to VMKP with at least two identical bins, and the $O(d)$ bound remains tight even in this setting.
• The analysis confirms that the algorithm’s performance is robust and optimal even under unbounded computational power, with no complexity assumptions.