Two uniform machines with nearly equal speeds: Unified approach to known sum and known optimum in semi on-line scheduling

György Dósa, M. Grazia Speranza, Zsolt Tuza

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We consider semi on-line scheduling on two uniform machines. The speed of the slow machine is normalized to 1 while the speed of the fast machine is assumed to be s ≥ 1. Jobs of size J1,J2, . . . arrive one at a time, and each Ji (i ≥ 1) has to be assigned to one of the machines before Ji+1 arrives. The assignment cannot be changed later. The processing time of the ith job is Ji on the slow machine and Ji/s on the fast one. The objective is to minimize the makes pan. We study both the case where the only information known in advance is the total sizeΣi≥1 Ji of the jobs and the case where the only information known in advance is the optimum makes pan. For each of these two cases, we almost completely determine the best possible competitive ratio of semi on-line algorithms compared to the off-line optimum, as a function of s in the range 1 ≤s < 1+ √ 17 4 ≈ 1.2808, except for a very short subinterval around s = 1.08. We also prove that the best competitive ratio achievable for known optimum is at least as good as the one for known sum, even for any number of uniform machines of any speeds.

Original languageEnglish
Pages (from-to)458-480
Number of pages23
JournalJournal of Combinatorial Optimization
Volume21
Issue number4
DOIs
Publication statusPublished - May 1 2011

Keywords

  • Competitive analysis
  • Semi on-line scheduling
  • Uniform machines

ASJC Scopus subject areas

  • Computer Science Applications
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Theory and Mathematics
  • Applied Mathematics

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