, Volume 55, Issue 1, pp 205–226 | Cite as

Scheduling on Unrelated Machines under Tree-Like Precedence Constraints

  • V. S. Anil KumarEmail author
  • Madhav V. Marathe
  • Srinivasan Parthasarathy
  • Aravind Srinivasan


We present polylogarithmic approximations for the R|prec|C max  and R|prec|∑ j w j C j problems, when the precedence constraints are “treelike”—i.e., when the undirected graph underlying the precedences is a forest. These are the first non-trivial generalizations of the job shop scheduling problem to scheduling with precedence constraints that are not just chains. These are also the first non-trivial results for the weighted completion time objective on unrelated machines with precedence constraints of any kind. We obtain improved bounds for the weighted completion time and flow time for the case of chains with restricted assignment—this generalizes the job shop problem to these objective functions. We use the same lower bound of “congestion + dilation”, as in other job shop scheduling approaches (e.g. Shmoys, Stein and Wein, SIAM J. Comput. 23, 617–632, 1994). The first step in our algorithm for the R|prec|C max  problem with treelike precedences involves using the algorithm of Lenstra, Shmoys and Tardos to obtain a processor assignment with the congestion + dilation value within a constant factor of the optimal. We then show how to generalize the random-delays technique of Leighton, Maggs and Rao to the case of trees. For the special case of chains, we show a dependent rounding technique which leads to a bicriteria approximation algorithm for minimizing the flow time, a notoriously hard objective function.


Precedence-constrained scheduling Job-shop scheduling Approximation algorithms Randomized algorithms 


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • V. S. Anil Kumar
    • 1
    Email author
  • Madhav V. Marathe
    • 1
  • Srinivasan Parthasarathy
    • 2
  • Aravind Srinivasan
    • 3
  1. 1.Virginia Bioinformatics Institute and Department of Computer ScienceVirginia Tech.BlacksburgUSA
  2. 2.IBM T.J. Watson Research CenterYorktown HeightsUSA
  3. 3.Department of Computer Science and Institute for Advanced Computer StudiesUniversity of MarylandCollege ParkUSA

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