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Improved Approximation Algorithms for Non-Preemptive Throughput Maximization

Alexander Armbruster, Fabrizio Grandoni, Antoine Tinguely, Andreas Wiese

Abstract

The (Non-Preemptive) Throughput Maximization problem is a natural and fundamental scheduling problem. We are given $n$ jobs, where each job $j$ is characterized by a processing time and a time window, contained in a global interval $[0,T)$, during which~$j$ can be scheduled. Our goal is to schedule the maximum possible number of jobs non-preemptively on a single machine, so that no two scheduled jobs are processed at the same time. This problem is known to be strongly NP-hard. The best-known approximation algorithm for it has an approximation ratio of $1/0.6448 + \varepsilon \approx 1.551 + \varepsilon$ [Im, Li, Moseley IPCO'17], improving on an earlier result in [Chuzhoy, Ostrovsky, Rabani FOCS'01]. In this paper we substantially improve the approximation factor for the problem to $4/3+\varepsilon$ for any constant~$\varepsilon>0$. Using pseudo-polynomial time $(nT)^{O(1)}$, we improve the factor even further to $5/4+\varepsilon$. Our results extend to the setting in which we are given an arbitrary number of (identical) machines.

Improved Approximation Algorithms for Non-Preemptive Throughput Maximization

Abstract

The (Non-Preemptive) Throughput Maximization problem is a natural and fundamental scheduling problem. We are given jobs, where each job is characterized by a processing time and a time window, contained in a global interval , during which~ can be scheduled. Our goal is to schedule the maximum possible number of jobs non-preemptively on a single machine, so that no two scheduled jobs are processed at the same time. This problem is known to be strongly NP-hard. The best-known approximation algorithm for it has an approximation ratio of [Im, Li, Moseley IPCO'17], improving on an earlier result in [Chuzhoy, Ostrovsky, Rabani FOCS'01]. In this paper we substantially improve the approximation factor for the problem to for any constant~. Using pseudo-polynomial time , we improve the factor even further to . Our results extend to the setting in which we are given an arbitrary number of (identical) machines.

Paper Structure

This paper contains 18 sections, 35 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Other related Work
  4. Polynomial time approximation algorithm
  5. Construction of blocks and superblocks
  6. The linear program
  7. Rounding algorithm
  8. Analysis
  9. Proof of Lemma \ref{['lem:fractional-matching-exists']}
  10. A pseudopolynomial time $(5/4+\varepsilon)$-approximation
  11. Solving the linear program and instances with small optimal solutions
  12. Alternative rounding algorithm
  13. Extension to multiple machines
  14. Construction of blocks and superblocks
  15. The linear program
  16. ...and 3 more sections

Key Result

Theorem 1

For any constant $\varepsilon>0$, there is a polynomial-time randomized $(4/3+\varepsilon)$-approximation algorithm for Throughput Maximization.

Figures (1)

  • Figure 1: The blue and red lines delimitate the blocks and superblocks, respectively. Job $j$ is global and can be scheduled within its boundary blocks $B_{j,L}$ and $B_{j,R}$ (marked blue) or within one of the two superblocks that it spans (marked red). It cannot be scheduled between a spanned superblock and $B_{j,L}$ or $B_{j,R}$, and cannot intersect two (or more) blocks. Job $j'$ is local. Its (unique) boundary block is marked in green.

Theorems & Definitions (61)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 51 more