Some Very Easy Knapsack / Partition Problems

Consider the problem of partitioning a group of b indistinguishable objects into subgroups, each of size at leastl and at most u. The objective is to minimize the additive separable cost of the partition, where the cost associated with a subgroup of size j is c(j). In the case that c(.) is convex, we show how to solve the problem in O(log u-/ + 1) steps. In the case that c(.) is concave, we solve the problem in O(min(/, b/u, (b/I)-(b/u), u-I)) steps. This problem generalizes a lot-sizing result of Chand and has potential applications in clustering. CONSIDER the problem of partitioning a group of b objects into subgroups, each of size at least and at most u, where 1 I < u < b, and 1, u, and b are integers. The objective is to minimize an additive cost Y7=' c(j)Xj where c(.) is some real-valued function and x is the number of subgroups of size j. This problem can be expressed as the following knapsack problem P: (P) Minimize j=1 c(j)xj subject to JY= 1 jxj = b xj > integer for As is well-known, the problem P can be solved in O((u-I)b) steps via a dynamic programming recursion. Moreover, if b u 2 then P can be solved in O((u-I)u) steps because the optimal solution to the associated group problem is feasible for P. (See Garfinkel and Nemhauser 1972; and Denardo and Fox 1979 for further details.) The purpose of this note is to provide very efficient algorithms for situations in which c(.) is either concave or convex. In particular, we show that we can solve versions of P in which c(.) is convex in O(log(u-I + 1)) steps. This algorithm extends a previous algorithm by Chand (1982) for a variant of the discrete time EOQ model, as mentioned below. If c(.) is concave, we show how to solve the knapsack/partition problem in O(min(l, b/u, (b/l)-(b/u), u-1)) steps. It is an open question as to Subject classification: 702 some very easy knapsack/partition problems.