# The Competitive Ratio of Threshold Policies for Online Unit-density Knapsack Problems

arXiv: https://arxiv.org/abs/1907.08735 · HTML: https://mlsanity.com/paper/1907.08735/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jul 2019.

## Built on by (most cited first, 4 of 4)

- [Near-Optimal Consistency-Robustness Trade-Offs for Learning-Augmented Online Knapsack Problems](https://mlsanity.com/paper/2406.18752/) (2024)
- [Learning-Augmented Competitive Algorithms for Spatiotemporal Online Allocation with Deadline Constraints](https://mlsanity.com/paper/2408.07831/) (2024)
- [Learning-Augmented Algorithms for Online Concave Packing and Convex Covering Problems](https://mlsanity.com/paper/2411.08332/) (2024)
- [A Simple Learning-Augmented Algorithm for Online Packing with Concave Objectives](https://mlsanity.com/paper/2406.03574/) (2024)

## Abstract (arXiv)

We study a wholesale supply chain ordering problem. In this problem, the supplier has an initial stock, and faces an unpredictable stream of incoming orders, making real-time decisions on whether to accept or reject each order. What makes this wholesale supply chain ordering problem special is its ``knapsack constraint,'' that is, we do not allow partially accepting an order or splitting an order. The objective is to maximize the utilized stock. We model this wholesale supply chain ordering problem as an online unit-density knapsack problem. We study randomized threshold algorithms that accept an item as long as its size exceeds the threshold. We derive two optimal threshold distributions, the first is 0.4324-competitive relative to the optimal offline integral packing, and the second is 0.4285-competitive relative to the optimal offline fractional packing. Both results require optimizing the cumulative distribution function of the random threshold, which are challenging infinite-dimensional optimization problems. We also consider the generalization to multiple knapsacks, where an arriving item has a different size in each knapsack. We derive a 0.2142-competitive algorithm for this problem. We also show that any randomized algorithm for this problem cannot be more than 0.4605-competitive. This is the first upper bound strictly less than 0.5, which implies the intrinsic challenge of knapsack constraint. We show how to naturally implement our optimal threshold distributions in the warehouses of a Latin American chain department store. We run simulations on their order data, which demonstrate the efficacy of our proposed algorithms.