Online Knapsack Problems with Estimates

2025-04-30

AI Summary

This paper studies the online activity selection problem under budget constraints with estimation errors: given a sequence of activities (e.g., academic conferences), each actual cost lies within ±δ of its estimated value, and irrevocable or limited-removal decisions must be made online. Addressing this realistic setting, we present the first online algorithms achieving tight optimal competitive ratios for arbitrary δ, rigorously characterizing the fundamental trade-off between estimation uncertainty and decision performance. Our approach integrates robust optimization, threshold-based policies, and a rescheduling mechanism. We derive provably optimal, unimprovable competitive ratios—both for the irremovable and the limited-removal variants—and provide explicit, constructive algorithms. These results bridge a critical gap between deterministic offline optimization and classical adversarial online models with no prior information.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationPlanning, Routing, and Scheduling: Scheduling under UncertaintyMachine Learning: Online Learning & Bandits

Application Category

Economics, Online Markets and Human Computation: Research challenges in human and human-AI computationGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Imagine you are a computer scientist who enjoys attending conferences or workshops within the year. Sadly, your travel budget is limited, so you must select a subset of events you can travel to. When you are aware of all possible events and their costs at the beginning of the year, you can select the subset of the possible events that maximizes your happiness and is within your budget. On the other hand, if you are blind about the options, you will likely have a hard time when trying to decide if you want to register somewhere or not, and will likely regret decisions you made in the future. These scenarios can be modeled by knapsack variants, either by an offline or an online problem. However, both scenarios are somewhat unrealistic: Usually, you will not know the exact costs of each workshop at the beginning of the year. The online version, however, is too pessimistic, as you might already know which options there are and how much they cost roughly. At some point, you have to decide whether to register for some workshop, but then you are aware of the conference fee and the flight and hotel prices. We model this problem within the setting of online knapsack problems with estimates: in the beginning, you receive a list of potential items with their estimated size as well as the accuracy of the estimates. Then, the items are revealed one by one in an online fashion with their actual size, and you need to decide whether to take one or not. In this article, we show a best-possible algorithm for each estimate accuracy $delta$ (i.e., when each actual item size can deviate by $pm delta$ from the announced size) for both the simple knapsack and the simple knapsack with removability.

Problem

• Online knapsack with estimated item sizes
• Decision-making under partial cost information
• Optimal algorithm for varying estimate accuracy

Innovation

• Online knapsack with estimated item sizes
• Algorithm adapts to estimate accuracy delta
• Supports both removable and non-removable items