Transformations on Regular Nondominated Coteries and Their Applications

A coterie under an underlying set U is a family of subsets of U such that every pair of subsets has at least one element in common, but neither is a subset of the other. A coterie C under U is said to be nondominated (ND) if there is no other coterie D under U such that, for every Q ∈ C, there exists Q ∈ D satisfying Q ⊆ Q. We introduce the operation σ which transforms a ND coterie to another ND coterie. A regular coterie is a natural generalization of a vote-assignable coterie. We show that any regular ND coterie C can be transformed to any other regular ND coterie D by judiciously applying the σ operation to C at most |C| + |D| − 2 times. As another application of the σ operation, we present an incrementally polynomial-time algorithm for generating all regular ND coteries. We then introduce the concept of a g-regular functional as a generalization of availability. We show how to construct an optimum coterie C with respect to a g-regular functional in O(n3|C|) time, where n = |U |. Finally, we discuss the structures of optimum coteries with respect to a g-regular functional.