Combinatorics of Mixed Graphs -- Complexity and Homomorphism

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Graph models are a fundamental tool for those working with relational data; they allow social scientists to study changing social and political alliances within a social network; they allow medical schools and hospitals a method to fairly pair medical residents with hospitals; and they allow engineers to find efficient schedules for large-scale engineering projects. However, these models are limited in that they often assume that the relationship between a pair of objects (such as people, resident and hospitals, or tasks) is identical, regardless of the pair being considered. In this research we study a graph model that allows for a more sophisticated notion of relationship. For example, using such a model, researchers can study social networks in which the relationship between family members is noted as a different kind of relationship as that between friends. The long-term goal of this research is to develop tools for graph models with this more sophisticated notion of relationship so that researchers and industry may apply these models in their respective areas. In particular, this research focusses on assignment problems; problems in which objects are grouped, or not grouped, based on their relationship to one another. The past study of assignment problems has lead to improvements in compiler use in parallel processing, in frequency assignments in radio communications networks, and in timetable scheduling. This research explores fundamental questions about the computational nature of assignment problems defined for graph models with more sophisticated notions of relationship. It characterizes these problems based on the existence (or suspected lack thereof) of efficient algorithms. For those problems for which it is expected that no efficient algorithm can exist, this research develops approximation algorithms, algorithms that provide a solution that is guaranteed to approximate the optimal solution within a fixed error bound.