Packing shelves with items that divide the shelves' length: A case of a universal number partition problem

Moshe Dror, James B. Orlin, Michael Zhu

Research output: Contribution to journalArticle

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Abstract

This note examines the likelihood of packing two identical one dimensional shelves of integer length L by items whose individual lengths are divisors of L, given that their combined length sums-up to 2L. We compute the number of packing failures and packing successes for integer shelve lengths L, 1 ≤ L ≤ 1000, by implementing a dynamic programming scheme using a problem specific "boundedness property". The computational results indicate that the likelihood of a packing failure is very rare. We observe that the existence of packing failures is tied to the number of divisors of L and prove that the number of divisors has to be at least 8 for a packing failure to exist.

Original languageEnglish (US)
Pages (from-to)189-198
Number of pages10
JournalDiscrete Mathematics, Algorithms and Applications
Volume2
Issue number2
DOIs
Publication statusPublished - Jun 1 2010

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Keywords

  • divisibility and packing
  • Universal Number Partition

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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