A Dense Hierarchy of Sublinear Time Approximation Schemes for Bin Packing

  • Richard Beigel
  • Bin Fu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7285)

Abstract

The bin packing problem is to find the minimum number of bins of size one to pack a list of items with sizes a 1,…, a n in (0,1]. Using uniform sampling, which selects a random element from the input list each time, we develop a randomized \(O({n(\log\log n)\over \sum_{i=1}^n a_i}+({1\over \epsilon})^{O({1\over\epsilon})})\) time (1 + ε)-approximation scheme for the bin packing problem. We show that every randomized algorithm with uniform random sampling needs \(\Omega({n\over \sum_{i=1}^n a_i})\) time to give an (1 + ε)-approximation. For each function s(n): N → N, define ∑ (s(n)) to be the set of all bin packing problems with the sum of item sizes equal to s(n). We show that ∑ (n b ) is NP-hard for every b ∈ (0,1]. This implies a dense sublinear time hierarchy of approximation schemes for a class of NP-hard problems, which are derived from the bin packing problem. We also show a randomized streaming approximation scheme for the bin packing problem such that it needs only constant updating time and constant space, and outputs an (1 + ε)-approximation in \(({1\over \epsilon})^{O({1\over\epsilon})}\) time. Let S(δ)-bin packing be the class of bin packing problems with each input item of size at least δ. This research also gives a natural example of NP-hard problem (S(δ)-bin packing) that has a constant time approximation scheme, and a constant time and space sliding window streaming approximation scheme, where δ is a positive constant.

Keywords

Approximation Scheme Minimum Span Tree Uniform Sampling Large Item Small Item 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Richard Beigel
    • 1
  • Bin Fu
    • 2
  1. 1.CIS DepartmentTemple UniversityPhiladelphiaUSA
  2. 2.Department of Computer ScienceUniversity of Texas-Pan AmericanEdinburgUSA

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