This dissertation focuses on the properties of the fairest semi-matchings in bipartite graphs. The previous works related to the semi-matching problem, mainly focus on the algorithms design to achieve one fairest semi-matching regarding one fairness measure. One big concern about the fairest semi-matchings is that whether they are always the fairest no matter which fairness measure is picked? This dissertation addresses this concern and proves the existence of a set of fairest semi-matching(s) which are universally agreed by all fairness measures. This work is our first main contribution.
Then we studied on the set of fairest semi-matchings regarding how all of them are related or what they have in common. The main contributions achieved are: given a bipartite graph, from one arbitrary fairest semi-matching (which is easy to achieve), we can understand some important attributes for the entire set of fairest semi-matchings: 1) the classification of the edges in the bipartite graph - whether each edge is used by all, none, or some of the fairest semi-matchings; 2) the partition of user and resource vertices in the bipartite graph - the allocating of all the fairest semi-matchings are all within the partitions, and each user vertex has a very narrow quota range (at most differ by 1, and is predictable from the knowledge gained from one fairest semi-matching) among all the fairest semi-matchings. This work is our second main contribution.
Furthermore, we consider the scenario of indivisible resources which indicates each resource can be split and allocated to maybe more than one user. In this work, we explore that the similarity between the set of fairest semi-matching and the set of fairest fractional allocations. And we conclude that the constant vertices partition across all the fairest semi-matchings is also constant across all the fairest fractional allocations - the allocating of all fairest semi-matchings and all the fairest fractional allocations are always within the partitions. Moreover, for each user, the difference of its quotas between one fairest semi-matching and one fairest fractional allocation is limited (either 0 or bound by 1) and predictable. This work is our third main contribution
