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Asymptotically Optimal Competitive Ratio for Online Allocation of Reusable Resources

Vineet Goyal, Garud Iyengar, Rajan Udwani

TL;DR

The performance guarantee for the algorithms is established by constructing a feasible solution to a novel system of inequalities that allows direct comparison with the clairvoyant benchmark instead of a linear programming (LP) relaxation of the benchmark.

Abstract

We consider the problem of online allocation (matching, budgeted allocations, and assortments) of reusable resources where an adversarial sequence of resource requests is revealed over time and any allocated resource is used/rented for a stochastic duration drawn independently from a resource dependent usage distribution. Previously, it was known that a greedy algorithm is 0.5--competitive against the clairvoyant benchmark that knows the entire sequence of requests in advance (Gong et al. (2021)). We give a novel algorithm that is $(1-1/e)$--competitive for arbitrary usage distributions when the starting capacity of each resource is large and the usage distributions are known. This is the best achievable competitive ratio guarantee for the problem, i.e., no online algorithm can have better competitive ratio. We also give a distribution oblivious online algorithm and show that it is $(1-1/e)$--competitive in special cases. At the heart of our algorithms is a new quantity that factors in the potential of reusability for each resource by (computationally) creating an asymmetry between identical units of the resource. We establish the performance guarantee for our algorithms by constructing a feasible solution to a novel system of inequalities that allows direct comparison with the clairvoyant benchmark instead of a linear programming (LP) relaxation of the benchmark. Our technique generalizes the primal-dual analysis framework for online resource allocation and may be of broader interest.

Asymptotically Optimal Competitive Ratio for Online Allocation of Reusable Resources

TL;DR

The performance guarantee for the algorithms is established by constructing a feasible solution to a novel system of inequalities that allows direct comparison with the clairvoyant benchmark instead of a linear programming (LP) relaxation of the benchmark.

Abstract

We consider the problem of online allocation (matching, budgeted allocations, and assortments) of reusable resources where an adversarial sequence of resource requests is revealed over time and any allocated resource is used/rented for a stochastic duration drawn independently from a resource dependent usage distribution. Previously, it was known that a greedy algorithm is 0.5--competitive against the clairvoyant benchmark that knows the entire sequence of requests in advance (Gong et al. (2021)). We give a novel algorithm that is --competitive for arbitrary usage distributions when the starting capacity of each resource is large and the usage distributions are known. This is the best achievable competitive ratio guarantee for the problem, i.e., no online algorithm can have better competitive ratio. We also give a distribution oblivious online algorithm and show that it is --competitive in special cases. At the heart of our algorithms is a new quantity that factors in the potential of reusability for each resource by (computationally) creating an asymmetry between identical units of the resource. We establish the performance guarantee for our algorithms by constructing a feasible solution to a novel system of inequalities that allows direct comparison with the clairvoyant benchmark instead of a linear programming (LP) relaxation of the benchmark. Our technique generalizes the primal-dual analysis framework for online resource allocation and may be of broader interest.

Paper Structure

This paper contains 49 sections, 56 theorems, 187 equations, 1 figure, 7 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Generalized Certificate of Optimality
  5. $\left(1-\frac{1}{e}\right)$ Competitive Algorithm for Arbitrary Usage Distributions
  6. The Random Process Viewpoint
  7. Useful Properties of $(F,\boldsymbol {\sigma},\boldsymbol{p})$ Random Process.
  8. Analysis of G-ALG and Sample G-ALG
  9. Faster and Distribution Oblivious Algorithms
  10. Analysis of RBA: Overview of Main Challenges
  11. Extensions to Online Assortments and Budgeted Allocation
  12. Numerical Experiments
  13. Experimental Setup
  14. Experimental Results
  15. Summary and Future Directions
  16. ...and 34 more sections

Key Result

Theorem 1

The competitive ratio of Balance with $g(x)=e^{-x}$ is less than $0.626$$(<(1-1/e))$ for OBMR.

Figures (1)

  • Figure 1: Comparison of Greedy, Balance, and RBA on Example \ref{['passivity2']}. Numbers on the $y$ axis denote the resource matched to $t_0$ w.h.p..

Theorems & Definitions (100)

  • Theorem 1
  • Theorem 2: Special Case of Main Result
  • Lemma 3
  • Lemma 4: Performance of G-ALG
  • Lemma 5: Monotonicity Property
  • Lemma 6: Adding Zero Probability Points
  • Lemma 7: Equivalence
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 90 more