Transformations on Regular Nondominated Coteries and Their Applications

Abstract

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 $Q \in C$, there exists $Q' \in D$ satisfying $Q' \subseteq Q$. We introduce the operation $\sigma$ 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 $\sigma$ operation to C at most |C|+|D|-2 times. As another application of the $\sigma$ 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.