On Maintaining Online Bipartite Matchings with Augmentations

In the thesis we focus on an algorithmic approach to one of many problems related to resource allocation. More precisely, its main result is a procedure for maintaining a maximum cardinality matching in a setting, where a part of the graph is revealed online, during the algorithm run. A matching M in a graph G = 〈V ,E〉 is any subset of edges M ⊆ E that are pairwise vertex-disjoint, i.e., each vertex has at most one incident edge in M. For example, consider a group of people interconnected by a symmetric relation of acquaintance. A matching then is a pairing of these people such that each person in a pair is acquainted with the other and nobody is in a more than one pair. We call M a maximum cardinality matching, or simply a maximum matching, if it is of biggest possible size, that is, for any other matching M ′ we have |M| > |M ′|. It is easiest to see the task of calculating maximum matching as a resource allocation problem in the bipartite case, that is, when the vertices V of G can be partitioned into two sets such that all the edges connect vertices from two different parts. Despite this limitation, practical applications of bipartite matching are numerous, especially, when each edge is labeled by weight that represents the benefit or cost associated with it. For example, medical students in the United States have been assigned to hospitals using a similar setting since the early 50’ of the last century. Even before, minimum weight matching was used to optimize problems motivated by transportation or classification of military personnel. Furthermore, both weighted and unweighted versions of the maximum matching problem found multiple uses as building blocks of more complex algorithms, having applications in mobile sensing systems, recommendation systems, algorithms including pattern recognition and several areas of bioinformatics, as well as in