Using dual approximation algorithms for scheduling problems: Theoretical and practical results

@article{Hochbaum1985UsingDA,
  title={Using dual approximation algorithms for scheduling problems: Theoretical and practical results},
  author={Dorit S. Hochbaum and David B. Shmoys},
  journal={26th Annual Symposium on Foundations of Computer Science (sfcs 1985)},
  year={1985},
  pages={79-89}
}
  • D. HochbaumD. Shmoys
  • Published 21 October 1985
  • Computer Science
  • 26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
The problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well-studied problem in the theory of approximation algorithms for NP-hard optimization problems. In this paper we present the strongest possible type of result for this problem, a polynomial approximation scheme. More precisely, for each ε, we give an algorithm that runs in time O((n/ε)1/ε2) and has relative error at most ε. For algorithms that are polynomial in n and m… 

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References

SHOWING 1-10 OF 17 REFERENCES

Processor scheduling with improved heuristic algorithms

The major effort of this research is devoted to proving that the MULTIFIT heuristic can be modified, without increasing its time complexity from O(NlogN), so that its worst-case performance bound is reduced from some value in the range {13/11, 6/5} to 72/61 times optimal, a better bound than that yielded by any other known polynomial-time algorithm.

A packing problem you can almost solve by sitting on your suitcase

The notion of a dual approximation algorithm, where for the bin-packing problem, the aim is to find approximate packings where at most the optimal number of bins are used, but the bins are allowed to be filled beyond their capacity, is introduced.

An efficient approximation scheme for the one-dimensional bin-packing problem

  • N. KarmarkarR. Karp
  • Computer Science
    23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
  • 1982
It is proved that the LP relaxation of bin packing, which was solved efficiently in practice by Gilmore and Gomory, has membership in P, despite the fact that it has an astronomically large number of variables.

Fast approximation algorithms for knapsack problems

  • E. Lawler
  • Computer Science
    18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
  • 1977
These algorithms are based on ideas of Ibarra and Kim, with modifications which yield better time and space bounds, and also tend to improve the practicality of the procedures.

Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems

An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* P)/P* < ~, where P* is the desired optimal sum.

Algorithms for Scheduling Independent Tasks

Three general techniques are presented to obtain approximate solutions for optimization problems solvable in this way, and polynomial time algorithms are applied to obtain “good” approximate solutions.

Bin packing can be solved within 1 + ε in linear time

For any listL ofn numbers in (0, 1) letL* denote the minimum number of unit capacity bins needed to pack the elements ofL. We prove that, for every positive ε, there exists anO(n)-time algorithmS

Tighter Bounds for the Multifit Processor Scheduling Algorithm

This paper considers the problem of nonpreemptively scheduling n independent jobs on m identical, parallel processors with the object of minimizing the “makespan”, or completion time for the entire processor, with the aim of minimize the "makespan" of the entire system.

Computers and Intractability: A Guide to the Theory of NP-Completeness

It is proved here that the number ofrules in any irredundant Horn knowledge base involving n propositional variables is at most n 0 1 times the minimum possible number of rules.

An Application of Bin-Packing to Multiprocessor Scheduling

This work considers one of the basic, well-studied problems of scheduling theory, that of nonpreemptively scheduling n independent tasks on m identical, parallel processors with the objective of minimizing the number of overlapping tasks.