Submodularity on Hypergraphs: From Sets to Sequences

TL;DR

This paper extends submodular functions to sequences using graphs and hypergraphs, introducing algorithms with provable guarantees for complex structures, and demonstrates their effectiveness in real-world applications like recommendations and course design.

Contribution

It introduces new algorithms for submodularity on sequences in general graphs and hypergraphs with bounded degrees, expanding beyond previous acyclic graph limitations.

Findings

Algorithms achieve constant factor approximations.

Effective in movie recommendation and online link prediction.

Applicable to designing course sequences for MOOCs.

Abstract

In a nutshell, submodular functions encode an intuitive notion of diminishing returns. As a result, submodularity appears in many important machine learning tasks such as feature selection and data summarization. Although there has been a large volume of work devoted to the study of submodular functions in recent years, the vast majority of this work has been focused on algorithms that output sets, not sequences. However, in many settings, the order in which we output items can be just as important as the items themselves. To extend the notion of submodularity to sequences, we use a directed graph on the items where the edges encode the additional value of selecting items in a particular order. Existing theory is limited to the case where this underlying graph is a directed acyclic graph. In this paper, we introduce two new algorithms that provably give constant factor approximations for general graphs and hypergraphs having bounded in or out degrees. Furthermore, we show the utility of our new algorithms for real-world applications in movie recommendation, online link prediction, and the design of course sequences for MOOCs.