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In this paper we address the problem of planning reliable landmarkbased robot navigation strategies in the presence of significant sensor uncertainty. The navigation environments are modeled with directed weighted graphs in which edges can be traversed with given probabilities. To construct robust and efficient navigation plans, we compute "expected shortest paths" in such graphs. We formulate the expected shortest paths problem as a Markov decision process and provide two algorithms for its solution. We demonstrate the practicality of our approach using an extensive experimental analysis using graphs with varying sizes and parameters.