### 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 = {u_{1}, u_{2}, . . ., u_{k}} together with a "size" v_{i} ≡ v(u_{i}) ∈ Z^{+}, such that v_{i} ≠ v_{j} for i ≠ j, a "frequency" a_{i} ≡ a(u_{i}) ∈ Z^{+}, and a positive integer (shelf length) L ∈ Z^{+} with the following conditions: (i) L = ∏^{n}_{j=1}p_{j}(p_{j} ∈ Z^{+} ∀j, p_{j} ≠ p_{l} for j ≠ l) and v_{i} = ∏ _{j∈Ai}p_{j}, A_{i} ⊆ {l, 2, . . ., n} for i = 1, . . ., n; (ii) (A_{i}\{{n-ary intersection}^{k}_{j=1}A_{j}}) ∩ (A_{l}\{{n-ary intersection}^{k}_{j=1}A_{j}}) = {circled division slash}∀i ≠ l. Note that v_{i}|L (divides L) for each i. If for a given m ∈ Z^{+}, ∑^{n}_{i=1}a_{i}v_{i} = 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 {b_{11}, b_{12}, . . ., b_{1m}, b_{21}, . . ., b_{n1}, . . ., b_{nm}}⊆ N such that ∑^{m}_{j=1}b_{ij} = a_{i}, i = 1, . . ., k, and ∑^{k}_{i=1}b_{ij}v_{i} = 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) |
---|---|

Pages (from-to) | 216-229 |

Number of pages | 14 |

Journal | Journal of Complexity |

Volume | 10 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1994 |

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### ASJC Scopus subject areas

- Computational Mathematics
- Analysis
- Mathematics (miscellaneous)

### Cite this

*Journal of Complexity*,

*10*(2), 216-229. https://doi.org/10.1006/jcom.1994.1010

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

Research output: Contribution to journal › Article

*Journal of Complexity*, vol. 10, no. 2, pp. 216-229. https://doi.org/10.1006/jcom.1994.1010

}

TY - JOUR

T1 - Utilizing Shelve Slots

T2 - Sufficiency Conditions for Some Easy Instances of Hard Problems

AU - Dror, Moshe

AU - Smith, Benjamin T.

AU - Trudeau, Martin

PY - 1994/6

Y1 - 1994/6

N2 - 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.

AB - 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.

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U2 - 10.1006/jcom.1994.1010

DO - 10.1006/jcom.1994.1010

M3 - Article

AN - SCOPUS:50749132002

VL - 10

SP - 216

EP - 229

JO - Journal of Complexity

JF - Journal of Complexity

SN - 0885-064X

IS - 2

ER -