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Approximating Vector Scheduling: Almost Matching Upper and Lower Bounds.

Nikhil Bansal1, Tim Oosterwijk2, Tjark Vredeveld2

  • 11Eindhoven University of Technology, Eindhoven, The Netherlands.

Algorithmica
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Summary
This summary is machine-generated.

This study addresses the Vector Scheduling problem, proving a double exponential time complexity is necessary for certain approximations. It introduces an efficient approximation algorithm, achieving the first EPTAS for this problem.

Keywords:
Integer Linear ProgramMultiprocessor ScheduleOptimum Integer SolutionResource AugmentationVector Schedule

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Area of Science:

  • Optimization
  • Computer Science
  • Algorithm Analysis

Background:

  • The Vector Scheduling problem generalizes makespan minimization to multi-dimensional resource allocation.
  • Existing approximation schemes for fixed dimensions have double exponential running times in the dimension 'd'.

Purpose of the Study:

  • To establish lower bounds on the computational complexity of Vector Scheduling.
  • To develop an improved, more efficient approximation algorithm.

Main Methods:

  • Proving inapproximability results based on the Exponential Time Hypothesis and subexponential time algorithms for NP-hard problems.
  • Developing a new approximation algorithm with a significantly improved running time.

Main Results:

  • Demonstrated that a double exponential dependence on dimension 'd' is necessary for certain approximation ratios.
  • Presented a new algorithm achieving an approximation ratio of (1+epsilon) with a running time of O(n * exp(poly(d/epsilon)))
  • Established lower bounds even with resource augmentation.

Conclusions:

  • The established lower bounds are essentially tight.
  • The developed algorithm provides the first Efficient Polynomial Time Approximation Scheme (EPTAS) for the Vector Scheduling problem.