Motivated by many real world applications such as recommen-dations in online shopping or entertainment, we consider theproblem of selecting sequences of items. In this paper we intro-duce a novel class of utility functions over sequences of items,strictly generalizing the commonly used class of submodularset functions. We encode the sequential dependencies betweenitems by a directed graph underlying the utility function. Clas-sical algorithms fail to achieve any constant factor approxi-mation guarantees on the problem of selecting sequences ofbounded length with maximum utility. We propose an efficientalgorithm for this problem that comes with strong theoreticalguarantees characterized by the structural properties of theunderlying graph. We demonstrate the effectiveness of ouralgorithm in synthetic and real world experiments on a movierecommendation dataset.

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