### 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) |
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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

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

### Cite this

*Journal of Complexity*,

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