Popular Condensation with two sided preferences and one sided ties

Many daily problems can be mapped into a graph matching problem and can be solved by using graph theory algorithms. There are many different definitions of optimization about graph matching. Consider optimization of matching on a bipartite graph, where the two partite sets represent sets of apprentices and teachers, respectively. Each apprentice has a preference list, ranking a nonempty subset of teachers in order of strict preference and each teacher put a nonempty subset of apprentices in a single tie as its preference list. An apprentice or a teacher a may prefer one matching over the other based on the people matched to a in the two matchings according to a's personal preference. A matching is said to be popular if there is no other matching that more vertices are better off in. Although some efficient algorithms have been proposed for finding a popular matching, a popular matching may not exist for those instances where the competition of some apprentices cannot be resolved. In this research, we consider an extension of the popular matching problem: the popular condensation problem. The popular condensation problem is to find a popular matching with the set of apprentices whose preferences are neglected, that is, condensing the instance to admit a local popular matching. For the best of its usage, we also want to find an optimal popular condensation that minimizes set of apprentices whose preferences are needed to be neglected.