Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

Moshe Dror, Benjamin T. Smith, Martin Trudeau

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u1, u2, …, uk} together with a “size” vi ∈ v(ui) ∈ Z+, such that vi ≠ vj for i ≠ j, a "frequency" ai ∈ a(ui) ∈ Z+, and a positive integer (shelf length) L ∈ Z+ with the following conditions: (i) L = ∏nj=1pj(pj ∈ Z+ ∀j, pj ≠ pl for j ≠ l) and vi = ∏ j∈Aipj, Ai ⊆ {l, 2, …, n} for i = 1, …, n; (ii) (Ai\{{n-ary intersection}kj=1Aj}) ⋂ (Al\{{n-ary intersection}kj=1Aj}) = {circled division slash}∀i ≠ l. Note that vi|L (divides L) for each i. If for a given m ∈ Z+, ∑ni=1aivi = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b11, b12, …, b1m, b21, …, bn1, …, bnm}⊆ N such that ∑mj=1bij = ai, i = 1, …, k, and ∑ki=1bijvi = L, j =1, …, m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.

Original languageEnglish (US)
Pages (from-to)216-229
Number of pages14
JournalJournal of Complexity
Volume10
Issue number2
DOIs
StatePublished - Jun 1994

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Statistics and Probability
  • Numerical Analysis
  • Mathematics(all)
  • Control and Optimization
  • Applied Mathematics

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