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

Moshe Dror, Benjamin T. Smith, Martin Trudeau

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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 language English (US) 216-229 14 Journal of Complexity 10 2 https://doi.org/10.1006/jcom.1994.1010 Published - Jun 1994

### Fingerprint

Sufficiency
Linear equations
Computational complexity
Polynomials
Linear Diophantine equation
Integer
Sufficient Conditions
Minimal Set
Divides
Division
Polynomial time
NP-complete problem
Intersection
Non-negative

### ASJC Scopus subject areas

• Computational Mathematics
• Analysis
• Mathematics (miscellaneous)

### Cite this

Utilizing Shelve Slots : Sufficiency Conditions for Some Easy Instances of Hard Problems. / Dror, Moshe; Smith, Benjamin T.; Trudeau, Martin.

In: Journal of Complexity, Vol. 10, No. 2, 06.1994, p. 216-229.

Research output: Contribution to journalArticle

Dror, Moshe ; Smith, Benjamin T. ; Trudeau, Martin. / Utilizing Shelve Slots : Sufficiency Conditions for Some Easy Instances of Hard Problems. In: Journal of Complexity. 1994 ; Vol. 10, No. 2. pp. 216-229.
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