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Approximation Algorithms for Machine Minimization

Authors Mohsen Mohammadi, Mohammad R. Salavatipour

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Subject Classification

ACM Subject Classification
  • Theory of computation → Scheduling algorithms
Keywords
  • Machine minimization
  • approximation algorithms
  • scheduling

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Abstract

In this paper we consider a classic scheduling problem known as Machine Minimization (MM). The input to MM is a set J of n jobs, where each job j has processing time p_j, release time r_j, and deadline d_j. The goal is to schedule the jobs to run non-preemptively on minimum number of machines such that each job is fully scheduled within its [r_j,d_j] interval and each machine runs at most one job at a time. This problem generalizes several NP-hard problems (e.g. the case of identical release time and deadlines reduces to the bin packing problem). Using Randomized Rounding [Prabhakar Raghavan and Clark D. Thompson, 1987] one can get an O((log n)/(log log n))-approximation. Chuzhoy et al. [Julia Chuzhoy and Paolo Codenotti, 2009] presented an algorithm that uses O(opt²) machines (i.e. O(opt)-approximation). Combined with the earlier work this yields an O(√{(log n)/(log log n)})-approximation and this remains the best known result for over 20 years. Even for when the ratio of largest to smallest processing time p_{max}/p_{min} is bounded, or the number of distinct processing times are bounded, there is no (universal) constant approximation. In this paper, we present a number of results. When p_max/p_min = ρ we present an algorithm that yields an asymptotic (2+ε)-approximation in time n^{O(ρ⁴/ε⁴)}. When the number of distinct processing times is c, we present a 2-approximation with run time n^{O(c²log³ n)}. If we have c distinct processing times and p_max/p_min = ρ we present an asymptotic (1+ε)-approximation that runs in time n^O(c²⋅ρ²⋅log³n/ε²).

Cite As Get BibTex

Mohsen Mohammadi and Mohammad R. Salavatipour. Approximation Algorithms for Machine Minimization. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 136:1-136:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.136

Author Details

Mohsen Mohammadi
  • Department of Computing Science, University of Alberta, Edmonton, Canada
Mohammad R. Salavatipour
  • Department of Computing Science, University of Alberta, Edmonton, Canada

Funding

  • Salavatipour, Mohammad R.: Supported by NSERC.

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