The classical bin packing problem can be stated as follows: We are given a multiset of items {a1, ..., an} with sizes in [0,1], and want to pack them into a minimum number of bins, each of which with capacity one. There are several applications of this problem, for example in the field of logistics: We can interpret the i-th item as time a package deliverer spends for the i-th tour. Package deliverers have a daily restricted working time, and we want to assign the tours such that the number of package deliverers needed is minimized. Another setup is to think of the items as boxes with a standardized basis, but variable height. Then, the goal is to pack these boxes into a container, which is standardized in all three dimensions. Moreover, applications of variants of the classical bin packing problem arise in cloud computing, when we have to store virtual machines on servers. Besides its practical relevance, the bin packing problem is one of the fundamental problems in theoretical computer science: It was proven many years ago that under standard complexity assumptions it is not possible to compute the value of an optimal packing of the items efficiently - classical bin packing is NP-complete. Computing the value efficiently means that the runtime of the algorithm is bounded polynomially in the number of items we have to pack. Besides the offline version, where we know all items at the beginning, also the online version is of interest: Here, the items are revealed one-by-one and have to be packed into a bin immediately and irrevocably without knowing which and how many items will still arrive in the future. Also this version is of practical relevance. In many situations we do not know the whole input at the beginning: For example we are unaware of the requirements of future virtual machines, which have to be stored, or suddenly some more packages have to be delivered, and some deliverers already started their tour. We can think of the classical theoretical analysis of an online algorithm A as follows: An adversary studies the behavior of the algorithm and afterwards constructs a sequence of items I. Then, the performance is measured by the number of used bins by A performing on I, divided by the value of an optimal packing of the items in I. The adversary tries to choose a worst-case sequence so this way to measure the performance is very pessimistic. Moreover, the chosen sequences I often turn out to be artificial: For example, in many cases the sizes of the items increase monotonically over time. Instances in practice are often subject to random influence and therefore it is likely that they are good-natured. In this thesis we analyze the performance of online algorithms with respect to two stochastic models. The first model is the following: The adversary chooses a set of items SI and a distribution F on SI. Then, the items are drawn independently and identically distributed according to F. In the second model the adversary chooses a finite set of items SI and then these items arrive in random order, that is random with respect to the uniform distribution on the set of all possible permutations of the items. It is possible to show that the adversary in the second stochastic model is at least as powerful as in the first one. We can classify the results in this thesis in three parts: In the first part we consider the complexity of classical bin packing and its variants cardinality-constrained and class-constrained bin packing in both stochastic models. That is, we determine if it is possible to construct algorithms that are in expectation nearly optimal for large instances that are constructed according to the stochastic models or if there exist non-trivial lower bounds. Among other things we show that the complexity of class-constrained bin packing differs in the two models under consideration. In the second part we deal with bounded-space bin packing and the dual maximization variant bin covering. We show that it is possible to overcome classical worst-case bounds in both models. In other words, we see that bounded-space algorithms benefit from randomized instances compared to the worst case. Finally, we consider selected heuristics for class-constrained bin packing and the corresponding maximization variant class-constrained bin covering. Here, we note that the different complexity of class-constrained bin packing with respect to the studied stochastic models observed in the first part is not only a theoretical phenomenon, but also takes place for many common algorithmic approaches. Interestingly, when we apply the same algorithmic ideas to class-constrained bin covering, we benefit from both types of randomization similarly. </ul
