Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time Fairness in Online Knapsack Problems

Adam Lechowicz, Rik Sengupta, Bo Sun, Shahin Kamali, Mohammad Hajiesmaili

TL;DR

We study time fairness in the online knapsack problem (OKP) and formalize conditional time-independent fairness (CTIF) to balance static and dynamic pricing. We show that strict time-independent fairness (TIF) is infeasible in OKP and propose CTIF parametrizations, including Pareto-optimal deterministic algorithms (BASE and ECT), randomization insights, and learning-augmented LA-ECT with provable consistency and robustness. The results establish Pareto-optimal trade-offs between fairness and competitiveness and demonstrate empirical gains from learning-augmented predictions, while randomization offers theoretical advantages that may underperform in practice. The work advances practical fair resource allocation in online settings and suggests directions for applying CTIF to related online problems and group-fairness notions.

Abstract

The online knapsack problem is a classic problem in the field of online algorithms. Its canonical version asks how to pack items of different values and weights arriving online into a capacity-limited knapsack so as to maximize the total value of the admitted items. Although optimal competitive algorithms are known for this problem, they may be fundamentally unfair, i.e., individual items may be treated inequitably in different ways. We formalize a practically-relevant notion of time fairness which effectively models a trade off between static and dynamic pricing in a motivating application such as cloud resource allocation, and show that existing algorithms perform poorly under this metric. We propose a parameterized deterministic algorithm where the parameter precisely captures the Pareto-optimal trade-off between fairness (static pricing) and competitiveness (dynamic pricing). We show that randomization is theoretically powerful enough to be simultaneously competitive and fair; however, it does not work well in experiments. To further improve the trade-off between fairness and competitiveness, we develop a nearly-optimal learning-augmented algorithm which is fair, consistent, and robust (competitive), showing substantial performance improvements in numerical experiments.

Time Fairness in Online Knapsack Problems

TL;DR

We study time fairness in the online knapsack problem (OKP) and formalize conditional time-independent fairness (CTIF) to balance static and dynamic pricing. We show that strict time-independent fairness (TIF) is infeasible in OKP and propose CTIF parametrizations, including Pareto-optimal deterministic algorithms (BASE and ECT), randomization insights, and learning-augmented LA-ECT with provable consistency and robustness. The results establish Pareto-optimal trade-offs between fairness and competitiveness and demonstrate empirical gains from learning-augmented predictions, while randomization offers theoretical advantages that may underperform in practice. The work advances practical fair resource allocation in online settings and suggests directions for applying CTIF to related online problems and group-fairness notions.

Abstract

The online knapsack problem is a classic problem in the field of online algorithms. Its canonical version asks how to pack items of different values and weights arriving online into a capacity-limited knapsack so as to maximize the total value of the admitted items. Although optimal competitive algorithms are known for this problem, they may be fundamentally unfair, i.e., individual items may be treated inequitably in different ways. We formalize a practically-relevant notion of time fairness which effectively models a trade off between static and dynamic pricing in a motivating application such as cloud resource allocation, and show that existing algorithms perform poorly under this metric. We propose a parameterized deterministic algorithm where the parameter precisely captures the Pareto-optimal trade-off between fairness (static pricing) and competitiveness (dynamic pricing). We show that randomization is theoretically powerful enough to be simultaneously competitive and fair; however, it does not work well in experiments. To further improve the trade-off between fairness and competitiveness, we develop a nearly-optimal learning-augmented algorithm which is fair, consistent, and robust (competitive), showing substantial performance improvements in numerical experiments.
Paper Structure (25 sections, 45 equations, 5 figures, 1 table)

This paper contains 25 sections, 45 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem & Preliminaries
  4. Time Fairness
  5. Time-Independent Fairness (TIF)
  6. Conditional Time-Independent Fairness (CTIF)
  7. Online Fair Algorithms
  8. Pareto-optimal deterministic algorithms
  9. Proof of Lemma \ref{['lem:add']}
  10. Case I: when $v \ge \frac{U}{1 + \alpha \beta'}$.
  11. Case II: when $L < v < \frac{U}{1 + \alpha \beta'}$.
  12. Case I.
  13. Case II.
  14. Randomization helps
  15. Prediction helps
  16. ...and 10 more sections

Figures (5)

  • Figure 1: Plotting the threshold functions for several $\mathsf{OKP}$ algorithms. (a)$\mathsf{ZCL}$ (§ \ref{['sec:prelims']}) and the baseline algorithm (see §\ref{['sec:fairness']}); (b)$\mathsf{ECT}$ (§\ref{['sec:deter']}); (c)$\mathsf{LA\text{-}ECT}$ (§\ref{['sec:preds']})
  • Figure 2: Trade-off for the baseline algorithm and the Pareto-optimal lower bound. $U/L = 5$.
  • Figure 3: $\mathcal{I}_x$ consists of $N_x$ batches of items, arriving in increasing order of value density.
  • Figure 4: Numerical experiments, with varying value density fluctuation $U/L$.
  • Figure 5: Learning-augmented experiments, with $U/L = 500$ and different prediction errors.

Theorems & Definitions (11)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 1 more