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Online Bin Covering with Frequency Predictions

Magnus Berg, Shahin Kamali

TL;DR

The paper addresses online bin covering when item sizes come from a fixed finite set $S$ by analyzing purely online performance, prediction-augmented approaches, and stochastic settings. It introduces the Group Covering (GC) framework to achieve near-optimal consistency under accurate predictions and develops robust hybrids that trade off consistency and robustness. In the stochastic regime, the authors leverage PAC-learnability of discrete distributions to design a purely online algorithm POPC$_{\varepsilon}^{\delta}$ that achieves near-optimal average-case performance with high probability, independent of the unknown distribution. Collectively, the work provides a unified treatment of online bin covering with predictions and learning, delivering provable guarantees across adversarial, prediction-accurate, and stochastic scenarios, with practical implications for online algorithms augmented by historical frequency data.

Abstract

We study the discrete bin covering problem where a multiset of items from a fixed set $S \subseteq (0,1]$ must be split into disjoint subsets while maximizing the number of subsets whose contents sum to at least $1$. We study the online discrete variant, where $S$ is finite, and items arrive sequentially. In the purely online setting, we show that the competitive ratios of best deterministic (and randomized) algorithms converge to $\frac{1}{2}$ for large $S$, similar to the continuous setting. Therefore, we consider the problem under the prediction setting, where algorithms may access a vector of frequencies predicting the frequency of items of each size in the instance. In this setting, we introduce a family of online algorithms that perform near-optimally when the predictions are correct. Further, we introduce a second family of more robust algorithms that presents a tradeoff between the performance guarantees when the predictions are perfect and when predictions are adversarial. Finally, we consider a stochastic setting where items are drawn independently from any fixed but unknown distribution of $S$. Using results from the PAC-learnability of probabilities in discrete distributions, we also introduce a purely online algorithm whose average-case performance is near-optimal with high probability for all finite sets $S$ and all distributions of $S$.

Online Bin Covering with Frequency Predictions

TL;DR

The paper addresses online bin covering when item sizes come from a fixed finite set by analyzing purely online performance, prediction-augmented approaches, and stochastic settings. It introduces the Group Covering (GC) framework to achieve near-optimal consistency under accurate predictions and develops robust hybrids that trade off consistency and robustness. In the stochastic regime, the authors leverage PAC-learnability of discrete distributions to design a purely online algorithm POPC that achieves near-optimal average-case performance with high probability, independent of the unknown distribution. Collectively, the work provides a unified treatment of online bin covering with predictions and learning, delivering provable guarantees across adversarial, prediction-accurate, and stochastic scenarios, with practical implications for online algorithms augmented by historical frequency data.

Abstract

We study the discrete bin covering problem where a multiset of items from a fixed set must be split into disjoint subsets while maximizing the number of subsets whose contents sum to at least . We study the online discrete variant, where is finite, and items arrive sequentially. In the purely online setting, we show that the competitive ratios of best deterministic (and randomized) algorithms converge to for large , similar to the continuous setting. Therefore, we consider the problem under the prediction setting, where algorithms may access a vector of frequencies predicting the frequency of items of each size in the instance. In this setting, we introduce a family of online algorithms that perform near-optimally when the predictions are correct. Further, we introduce a second family of more robust algorithms that presents a tradeoff between the performance guarantees when the predictions are perfect and when predictions are adversarial. Finally, we consider a stochastic setting where items are drawn independently from any fixed but unknown distribution of . Using results from the PAC-learnability of probabilities in discrete distributions, we also introduce a purely online algorithm whose average-case performance is near-optimal with high probability for all finite sets and all distributions of .
Paper Structure (20 sections, 12 theorems, 72 equations, 4 algorithms)

This paper contains 20 sections, 12 theorems, 72 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Contribution
  4. Preliminaries
  5. Online Discrete Bin Covering
  6. Performance Measures
  7. Predictions Setting
  8. A Consistency-Robustness Trade-Off for $\textsc{DBC}_{\mathcal{S}_k}^{\mathcal{F}}\xspace$
  9. A Near-Optimally Consistent Algorithm for $\textsc{DBC}_{S}^{\mathcal{F}}\xspace$
  10. The Strategy of Group Covering
  11. Analysis of $\textsc{GC}_{\varepsilon}\xspace$
  12. Robustifying $\textsc{GC}_{\varepsilon}\xspace$
  13. Bounding the Performance of $\textsc{Opt}\xspace$
  14. Bound on the Performance of $\textsc{GC}_{\varepsilon}\xspace$
  15. A Trust-Parametrized Family of Hybrid Algorithms
  16. ...and 5 more sections

Key Result

Theorem 2

Any $(1-\alpha)$-consistent deterministic algorithm for $\textsc{DBC}_{k}^{\mathcal{F}}\xspace$ is at most $2\alpha$-robust.

Theorems & Definitions (22)

  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Lemma 7
  • proof
  • ...and 12 more