Listing all delta partitions of a given set: Algorithm design and results

Abstract

Let α \alpha be a set of n n elements and δ \delta be a nonnegative integer. A δ \delta -partition of α \alpha is a set of pairwise disjoint nonempty subsets of α \alpha such that the union of the subsets is equal to α \alpha and every subset has a size greater than δ \delta . We formulate an algorithm for computing all δ \delta -partitions of a given n n -element set and show that the algorithm runs in O ( n ) {\mathcal{O}}\left(n) space and O ( n ) {\mathcal{O}}\left(n) delay time between any two successive outputs of δ \delta -partitions of the given set. An application of the notion of δ \delta -partitions is illustrated in the following scheduling problem. Suppose a factory has n n machines and m ≤ n m\le n jobs to complete daily. Every job can be accomplished by operating at least δ + 1 \delta +1 machines. A machine cannot work on multiple jobs simultaneously. According to a utilization policy of the factory’s management, no machine is allowed to be idle, so all machines should be running on some job. Find a daily schedule of the factory’s machines satisfying all the mentioned constraints. Let α \alpha be the set of the factory’s machines. Then, an α \alpha ’s δ \delta -partition with m m subsets is a legal schedule if every subset (in the δ \delta -partition) includes exclusively δ + 1 \delta +1 or more machines that run on the same job.