Extensions of secretary problems towards submodular objectives and temporal arrival

In online algorithms, a decider has to make irrevocable decisions before she has received all relevant information about a problem. This is relevant in a lot of practical applications, e.g., in routing or scheduling problems. Online algorithms are often analyzed as a game between the algorithm and a malicious adversary. The adversary designs the input instance in such a way that the online algorithm performs badly. This type of worst-case analysis perspective is overly pessimistic in practice. Therefore, we develop algorithms for problems where the adversary only fixes the problem instance but not the order in which it is revealed to the algorithm. Instead, we consider problems, where the order of arrival is uniformly at random. This type of problem is also known as secretary problems. Throughout this thesis, the online data represents items that the online algorithms can either select or reject. We evaluate our algorithms with competitive analysis, which is a comparison between the expected outcome of the algorithm and the optimal offline solution to the problem, and achieve improved guarantees compared to the literature. In the first technical part of this thesis, we consider submodular secretary problems. Here, the adversary specifies a submodular objective function on a set of items and some combinatorial constraints. The order in which the items are revealed to the algorithm is uniformly at random. The objective is to select items online such that the objective function is maximized. The second technical part is on the temp secretary problem. In this part, the objective function is linear and items arrive uniformly distributed over a time interval. Additionally to the relative order of the items from the random order model, this model also contains exact arrival dates, which allows for temporal constraints. At every point in time, the set of all objects selected within a given time horizon has to fulfill the constraint. The algorithms for both types of problems are closely related. Whenever a new decision is due, the online algorithm computes an offline optimal solution on all information that is available up to this point and uses this to guide the decision. Before we go into the technical details of the two types of problems considered, we give an introduction to the algorithmic ideas and relevant proof techniques. As a novel quality measure for online algorithms, we introduce the notion of α · c-competitive algorithms. Here, c is the classical in-expectation competitive ratio and α is the approximation guarantee of an algorithm for the offline variant of the problem. If computational power is no concern, then α is 1. Otherwise, if the computation in an online round has to obey space or time requirements, then α is the approximation ratio of an offline algorithm that fulfills these constraints. This gives a clean separation of the computational hardness from the information-theoretical hardness of the problems. This is specially relevant for online algorithms with submodular objectives, as offline maximization of a submodular function is NP-hard.