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

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∈ A i p j , A i ⊆ {l, 2, . , n} for i = 1, . , n; (ii) ( A i \\{⋂ k j=1 A j }) ∩ ( A l \\{⋂ k j=1 A j }) = ⊘∀ i ≠ l. Note that v i | L (divides L) for each i. If for a given m ∈ Z +, ∑ n i=1 a iv 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 1 m , b 21, . , b n1 , . , b nm }⊆ N such that ∑ m j=1 b ij = a i , i = 1, . , k, and ∑ k i=1 b ijv 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.