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Abstract. It is shown that the problem of maintaining the topological order of the nodes of a directed acyclic graph while inserting m edges can be solved in O(min{m 3/2 log n, m 3/2 + n 2 log n}) time, an improvement over the best known result of O(mn). In addition, we analyze the complexity of the same algorithm with respect to the treewidth k of the underlying undirected graph. We show that the algorithm runs in time O(mk log 2 n) for general k and that it can be implemented to run in O(n log n) time on trees, which is optimal. If the input contains cycles, the algorithm detects this.

Abstract. We define and study two versions of the bipartite matching problem in the framework of two-stage stochastic optimization with recourse. In one version the uncertainty is in the second stage costs of the edges, in the other version the uncertainty is in the set of vertices that needs to be matched. We prove lower bounds, and analyze efficient strategies for both cases. These problems model real-life stochastic integral planning problems such as commodity trading, reservation systems and scheduling under uncertainty.

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a b s t r a c tWe define and study two versions of the bipartite matching problem in the framework of two-stage stochastic optimization with recourse. In one version, the uncertainty is in the second stage costs of the edges, and in the other version, the uncertainty is in the set of vertices that needs to be matched. We prove lower bounds, and analyze efficient strategies for both cases. These problems model real-life stochastic integral planning problems, such as commodity trading, reservation systems and scheduling under uncertainty.