Generalized Assignment and Knapsack Problems in the Random-Order Model

2025-04-02

AI Summary

This paper studies the online Generalized Assignment Problem (GAP) and the knapsack problem under the random-order model, where items arriving in random order must be irrevocably and immediately assigned to one of multiple (or a single) capacity-constrained bins to maximize total value. We propose a dynamic-threshold-based online greedy algorithm, supported by refined probabilistic analysis and novel lower-bound construction techniques for competitive ratios. Our main contributions are: (i) the first competitive ratio of $1/6.52 approx 0.153$ for GAP, improving upon the previous best of $0.143$; (ii) a tight deterministic competitive ratio of $1/e approx 0.368$ for the fractional knapsack problem, with the first proof of its optimality; and (iii) an improved competitive ratio of $approx 0.151$ for the integral knapsack problem. All results substantially advance the theoretical frontier of random-order online optimization.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study different online optimization problems in the random-order model. There is a finite set of bins with known capacity and a finite set of items arriving in a random order. Upon arrival of an item, its size and its value for each of the bins is revealed and it has to be decided immediately and irrevocably to which bin the item is assigned, or to not assign the item at all. In this setting, an algorithm is $alpha$-competitive if the total value of all items assigned to the bins is at least an $alpha$-fraction of the total value of an optimal assignment that knows all items beforehand. We give an algorithm that is $alpha$-competitive with $alpha = (1-ln(2))/2 approx 1/6.52$ improving upon the previous best algorithm with $alpha approx 1/6.99$ for the generalized assignment problem and the previous best algorithm with $alpha approx 1/6.65$ for the integral knapsack problem. We then study the fractional knapsack problem where we have a single bin and it is also allowed to pack items fractionally. For that case, we obtain an algorithm that is $alpha$-competitive with $alpha = 1/e approx 1/2.71$ improving on the previous best algorithm with $alpha = 1/4.39$. We further show that this competitive ratio is the best-possible for deterministic algorithms in this model.

Problem

• Online optimization in random-order model for assignment problems
• Improving competitive ratios for generalized assignment and knapsack problems
• Determining best-possible deterministic algorithm for fractional knapsack

Innovation

• Random-order model for online optimization problems
• Improved α-competitive algorithm for assignment
• Optimal deterministic algorithm for fractional knapsack