Modifying an Instance of the Super-Stable Matching Problem
Naoyuki Kamiyama
TL;DR
This paper proves that if the problem of modifying an instance of the super-stable matching problem by deleting some bounded number of agents in such a way that there exists a super-stable matching in the modified instance, then the problem can be solved in polynomial time.
Abstract
Super-stability is one of the stability concepts in the stable matching problem with ties. It is known that there may not exist a super-stable matching, and the existence of a super-stable matching can be checked in polynomial time. In this paper, we consider the problem of modifying an instance of the super-stable matching problem by deleting some bounded number of agents in such a way that there exists a super-stable matching in the modified instance. First, we prove that if we are allowed to delete agents on only one side, then our problem can be solved in polynomial time. Interestingly, this result is obtained by carefully observing the existing algorithm for checking the existence of a super-stable matching. In addition, we prove that if we are allowed to delete agents on both sides, then our problem is NP-complete.