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Minimizing Makespan in Sublinear Time via Weighted Random Sampling

Bin Fu, Yumei Huo, Hairong Zhao

TL;DR

The authors address parallel-machine makespan minimization for large inputs by developing sublinear-time approximation algorithms based on weighted random sampling. The core idea is to construct a compact sketch of the input that preserves the critical distribution of job sizes, then solve the problem on this sketch using existing approximation schemes. They present two sublinear-time algorithms, one for known $n$ and one for unknown $n$, both achieving a $(1+\epsilon)$-type approximation that translates into a $(1+3\epsilon)$-approximate makespan on the full instance when combined with a sketch-to-schedule construction. The sketch framework and adaptive sampling provide guarantees with high probability and yield a sketch schedule that can generate a real schedule when full data becomes available. The results enable fast decision-making in large-scale scheduling scenarios and illustrate the power of weighted sampling in sublinear optimization.

Abstract

We consider the classical makespan minimization scheduling problem where $n$ jobs must be scheduled on $m$ identical machines. Using weighted random sampling, we developed two sublinear time approximation schemes: one for the case where $n$ is known and the other for the case where $n$ is unknown. Both algorithms not only give a $(1+3ε)$-approximation to the optimal makespan but also generate a sketch schedule. Our first algorithm, which targets the case where $n$ is known and draws samples in a single round under weighted random sampling, has a running time of $\tilde{O}(\tfrac{m^5}{ε^4} \sqrt{n}+A(\ceiling{m\over ε}, ε ))$, where $A(\mathcal{N}, α)$ is the time complexity of any $(1+α)$-approximation scheme for the makespan minimization of $\mathcal{N}$ jobs. The second algorithm addresses the case where $n$ is unknown. It uses adaptive weighted random sampling, %\textit{that is}, it draws samples in several rounds, adjusting the number of samples after each round, and runs in sublinear time $\tilde{O}\left( \tfrac{m^5} {ε^4} \sqrt{n} + A(\ceiling{m\over ε}, ε )\right)$. We also provide an implementation that generates a weighted random sample using $O(\log n)$ uniform random samples.

Minimizing Makespan in Sublinear Time via Weighted Random Sampling

TL;DR

The authors address parallel-machine makespan minimization for large inputs by developing sublinear-time approximation algorithms based on weighted random sampling. The core idea is to construct a compact sketch of the input that preserves the critical distribution of job sizes, then solve the problem on this sketch using existing approximation schemes. They present two sublinear-time algorithms, one for known and one for unknown , both achieving a -type approximation that translates into a -approximate makespan on the full instance when combined with a sketch-to-schedule construction. The sketch framework and adaptive sampling provide guarantees with high probability and yield a sketch schedule that can generate a real schedule when full data becomes available. The results enable fast decision-making in large-scale scheduling scenarios and illustrate the power of weighted sampling in sublinear optimization.

Abstract

We consider the classical makespan minimization scheduling problem where jobs must be scheduled on identical machines. Using weighted random sampling, we developed two sublinear time approximation schemes: one for the case where is known and the other for the case where is unknown. Both algorithms not only give a -approximation to the optimal makespan but also generate a sketch schedule. Our first algorithm, which targets the case where is known and draws samples in a single round under weighted random sampling, has a running time of , where is the time complexity of any -approximation scheme for the makespan minimization of jobs. The second algorithm addresses the case where is unknown. It uses adaptive weighted random sampling, %\textit{that is}, it draws samples in several rounds, adjusting the number of samples after each round, and runs in sublinear time . We also provide an implementation that generates a weighted random sample using uniform random samples.
Paper Structure (21 sections, 31 theorems, 70 equations)

This paper contains 21 sections, 31 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. New contribution
  4. Definitions and the Framework of Our sublinear Time Algorithms
  5. Definitions
  6. The Framework of the Sublinear Time Algorithms
  7. Sublinear Time Approximation Schemes for a Sketch Instance
  8. Sketch Schedule
  9. Sublinear Time Algorithms for Obtaining the Sketch Instance
  10. Preliminaries
  11. Generating a Sketch When $n$ is Known
  12. Generating a Sketch When $n$ is Unknown
  13. $\sum_{p_j > w_0 } p_j \le \delta OPT$ and $h = O(\log {m n})$
  14. Intervals in $H_1$: $\tilde{n}_k$ as an Approximation of ${n}_k$
  15. Intervals in $H_2$: $\tilde{n}_k$ as an Approximation of ${n}_k$
  16. ...and 6 more sections

Key Result

Theorem 1

When the number of jobs $n$ is known, using non-adaptive weighted sampling, there is a randomized $(1+\epsilon)$-approximation algorithm for the makespan minimization problem that runs in $\tilde{O}(\tfrac{m^5}{\epsilon^4} \sqrt{n} + A(\left\lceil m\over \epsilon\right\rceil, {\epsilon} ) )$ time.

Theorems & Definitions (35)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Definition 3
  • Proposition 1
  • Theorem 3
  • Definition 4
  • Theorem 4
  • Theorem 5
  • ...and 25 more