Scheduling on unrelated machines under tree-like precedence constraints

Authors:
V. S. Anil Kumar;Madhav V. Marathe;Srinivasan Parthasarathy;Aravind Srinivasan
Affiliations:
Basic and Applied Simulation Science (CCS-DSS), Los Alamos National Laboratory, MS M997, Los Alamos, NM;Virginia Bio-informatics Institute, and Department of Computer Science, Virginia Tech, Blacksburg;Department of Computer Science, University of Maryland, College Park, MD;Department of Computer Science, University of Maryland, College Park, MD
Venue:
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Year:
2005

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Abstract

We present polylogarithmic approximations for the R|prec|Cmax and R|prec|∑jwjCj problems, when the precedence constraints are “treelike” – i.e., when the undirected graph underlying the precedences is a forest. We also 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. The first step in our algorithm for the R|prec|Cmax 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 weighted completion time, we show a certain type of reduction to the makespan problem, which dovetails well with the lower bound we employ for the makespan problem. For the special case of chains, we show a dependent rounding technique which leads to improved bounds on the weighted completion time and new bicriteria bounds for the flow time.