Online optimization in the random-order model

In an online problem, information is revealed incrementally and decisions have to be made before the full information is known. This occurs in various applications like, for example, resource allocation or online ad assignment. To analyze the performance of algorithms for online problems, it is classically assumed that there is a malicious adversary who always provides the worst-possible input. This, however, is a very pessimistic assumption. Therefore, in recent years, a lot of research has been done to analyze input models where the power of the adversary is restricted. In this thesis, we consider online optimization problems in the random-order model. In this online model, an adversary specifies an input instance in advance but, in contrast to the classic model, he may not determine the order in which it is revealed to the algorithm. Instead, the input sequence is revealed in random order. We analyze several combinatorial generalizations of the famous secretary problem and present algorithms with improved competitive ratios for each of them. Specifically, the problems considered here are of packing type, namely, bipartite matching, combinatorial auctions, generalized assignment and packing linear programs. First, we analyze the edge-weighted bipartite matching problem where the vertices of one side arrive online in random order. For this problem, we give a surprisingly simple algorithm that generalizes the classic algorithm for the secretary problem. Since its expected competitive ratio matches the best-possible one for the secretary problem, the algorithm is optimal. The result also gives the best-possible competitive ratio for the matroid secretary problem on transversal matroids. Then, we present improved competitive ratios for combinatorial auctions with online bidders arriving in random order. They are generalizations of the weighted matching problem and we analyze various types of valuation functions. Namely, we consider auctions where the bidders are interested in bundles of bounded or unbounded cardinality or where the valuation functions are submodular. For the online generalized assignment problem, which is another generalization of the weighted matching problem, we present the first constant-competitive algorithm. This result also improves on the best known competitive ratio for the online knapsack problem. Finally, we consider online packing LPs where the variables are revealed online in random order. For these, we present an algorithm that obtains the best-possible competitive ratio on instances with high capacity ratio, i. e., where, for every row, the capacity is large compared to the maximum entry in the constraint matrix. Furthermore, this algorithm also gives close-to-optimal results when the capacity ratio is only bounded by a constant. Additionally, we show how to modify the algorithm in the presence of strategic agents to obtain a truthful mechanism with almost identical competitive ratio.