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In a nutshell, submodular functions encode an intuitive notion of diminishingreturns. As a result, submodularity appears in many important machine learningtasks such as feature selection and data summarization. Although there has beena large volume of work devoted to the study of submodular functions in recentyears, the vast majority of this work has been focused on algorithms thatoutput sets, not sequences. However, in many settings, the order in which weoutput items can be just as important as the items themselves.  To extend the notion of submodularity to sequences, we use a directed graphon the items where the edges encode the additional value of selecting items ina particular order. Existing theory is limited to the case where thisunderlying graph is a directed acyclic graph. In this paper, we introduce twonew algorithms that provably give constant factor approximations for generalgraphs and hypergraphs having bounded in or out degrees. Furthermore, we showthe utility of our new algorithms for real-world applications in movierecommendation, online link prediction, and the design of course sequences forMOOCs.

Submodularity on Hypergraphs: From Sets to Sequences