Journal of Combinatorial Optimization

, Volume 36, Issue 2, pp 508–531 | Cite as

Min-Sum Bin Packing

  • Leah Epstein
  • David S. Johnson
  • Asaf LevinEmail author


We study min-sum bin packing (MSBP). This is a bin packing problem, where the cost of an item is the index of the bin into which it is packed. The problem is equivalent to a batch scheduling problem we define, where the total completion time is to be minimized. The problem is NP-hard in the strong sense. We show that it is not harder than this by designing a polynomial time approximation scheme for it. We also show that several natural algorithms which are based on well-known bin packing heuristics (such as First Fit Decreasing) fail to achieve an asymptotic finite approximation ratio, whereas Next Fit Increasing has an absolute approximation ratio of at most 2, and an asymptotic approximation ratio of at most 1.6188. We design a new heuristic that applies Next Fit Increasing on the relatively small items and adds the larger items using First Fit Decreasing, and show that its asymptotic approximation ratio is at most 1.5604.


Bin packing Approximation algorithms Analysis of algorithms 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.AT&T Labs - ResearchNJUSA
  3. 3.Faculty of Industrial Engineering and ManagementThe TechnionHaifaIsrael

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