We consider several subclasses of the problem of grouping n items (indexed 1, 2, …, n) into m subsets so as to minimize the function g(S1, …, Sm). In general, these problems are very difficult to solve to optimality, even for the case m = 2. We provide several sufficient conditions on g(·) that guarantee that there is an optimum partition in which each subset consists of consecutive integers (or else the partition S1, …, Sm satisfies a more general condition called “semiconsecutiveness”). Moreover, by restricting attention to “consecutive” (or “semiconsecutive”) partitions, we can solve the partition problem in polynomial time for small values of m. If, in addition, g is symmetric, then the partition problem is solvable in purely polynomial time. We apply these results to generalizations of a problem in inventory groupings considered by the authors in a previous paper. We also relate the results to the Neyman-Pearson lemma in statistical hypothesis testing and to a graph partitioning problem of Barnes and Hoffman.