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Computational bounds on randomized algorithms for online bin stretching

Antoine Lhomme, Nicolas Catusse, Nadia Brauner

TL;DR

The paper investigates online bin stretching under randomized strategies, formulating the interaction as a request-answer game and applying a Stengel-inspired linear programming method to compute lower bounds on the best randomized online algorithms. By discretizing item sizes and solving finite LPs, the authors prove convergence of the lower bounds to the optimal randomized competitive ratio for the bin-stretching problem and operationalize a computational search to construct randomized upper bounds. In the two-bin case, they report a new $5/4$-competitive randomized algorithm, improving over the deterministic $4/3$ benchmark, and establish a $29/24$ lower bound for a restricted $\mathcal{M}_2$ class, illustrating both the promise and the current limits of the approach. The work also discusses theoretical limits of the method and provides directions for future improvements, including pruning strategies and extensions to related online optimization problems and predictions-based settings.

Abstract

A frequently studied performance measure in online optimization is competitive analysis. It corresponds to the worst-case ratio, over all possible inputs of an algorithm, between the performance of the algorithm and the optimal offline performance. However, this analysis may be too pessimistic to give valuable insight on a problem. Several workarounds exist, such as randomized algorithms. This paper aims to propose computational methods to construct randomized algorithms and to bound their performance on the classical online bin stretching problem. A game theory method is adapted to construct lower bounds on the performance of randomized online algorithms via linear programming. Another computational method is then proposed to construct randomized algorithms which perform better than the best deterministic algorithms known. Finally, another lower bound method for a restricted class of randomized algorithm for this problem is proposed.

Computational bounds on randomized algorithms for online bin stretching

TL;DR

The paper investigates online bin stretching under randomized strategies, formulating the interaction as a request-answer game and applying a Stengel-inspired linear programming method to compute lower bounds on the best randomized online algorithms. By discretizing item sizes and solving finite LPs, the authors prove convergence of the lower bounds to the optimal randomized competitive ratio for the bin-stretching problem and operationalize a computational search to construct randomized upper bounds. In the two-bin case, they report a new -competitive randomized algorithm, improving over the deterministic benchmark, and establish a lower bound for a restricted class, illustrating both the promise and the current limits of the approach. The work also discusses theoretical limits of the method and provides directions for future improvements, including pruning strategies and extensions to related online optimization problems and predictions-based settings.

Abstract

A frequently studied performance measure in online optimization is competitive analysis. It corresponds to the worst-case ratio, over all possible inputs of an algorithm, between the performance of the algorithm and the optimal offline performance. However, this analysis may be too pessimistic to give valuable insight on a problem. Several workarounds exist, such as randomized algorithms. This paper aims to propose computational methods to construct randomized algorithms and to bound their performance on the classical online bin stretching problem. A game theory method is adapted to construct lower bounds on the performance of randomized online algorithms via linear programming. Another computational method is then proposed to construct randomized algorithms which perform better than the best deterministic algorithms known. Finally, another lower bound method for a restricted class of randomized algorithm for this problem is proposed.
Paper Structure (13 sections, 5 theorems, 56 equations, 8 figures, 1 algorithm)

This paper contains 13 sections, 5 theorems, 56 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Online bin stretching
  2. Randomized algorithms
  3. Motivation
  4. The request-answer framework
  5. Lower bounds for online randomized algorithms with linear programming
  6. Bounds on randomized algorithms for online bin stretching
  7. Lower bounds with the linear programming method
  8. Finding randomized algorithms through computational searches
  9. Lower bounds on the performance of the $\mathcal{M}_2$ class
  10. A theoretical limit of the computational approach
  11. Conclusion and perspectives
  12. On the definition of randomized online algorithms
  13. Proof of convergence of the linear program for online bin stretching

Key Result

Theorem 4.1

For $m=2$ bins, there is a lower bound on the competitive ratio of randomized algorithm of $\frac{7}{6} \approx 1.1667$. For $m=3$, there is a lower bound of $\frac{19}{16} = 1.1875$, and for $m=4$, of $\frac{11}{9} \approx 1.2222$.

Figures (8)

  • Figure 1: Best known bounds on the minimal stretching factor of deterministic algorithms for the online bin stretching problem. Solid squares and circles represent bounds that were found through computational searches.
  • Figure 2: Adversarial strategy showing the lower bound of $\frac{4}{3}$ for the online bin stretching problem where 2 bins are available. In parentheses: occupied volume in each bin.
  • Figure 3: Two player zero-sum game example. Diamond nodes represent a decision node for the algorithm player (aiming to minimize the pay-off) and filled circles are for the adversary player (aiming to maximize the pay-off). S represents "Snow" for the adversary and "Skiing" for the algorithm, while C represents "Clear weather" for the adversary and "Chocolate" for the algorithm.
  • Figure 4: Two player zero-sum game example. Diamond nodes represent a decision node for the algorithm player and filled circles are for the adversary player. Variables $x$ and $y$ have been added to represent probabilities for possible moves.
  • Figure 5: Illustration of the game tree for constructing upper bounds in the deterministic case. In parentheses, the load of each bin. Circle nodes correspond to game states where the algorithm has to take a decision.
  • ...and 3 more figures

Theorems & Definitions (16)

  • Definition 2.1: Request answer game
  • Remark
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Theorem 4.1
  • Remark
  • Theorem 4.2
  • Theorem 4.3
  • ...and 6 more