Resolving the Approximability of Offline and Online Non-monotone DR-Submodular Maximization over General Convex Sets

Abstract

In recent years, maximization of DR-submodular continuous functions became an important research field, with many real-worlds applications in the domains of machine learning, communication systems, operation research and economics. Most of the works in this field study maximization subject to down-closed convex set constraints due to an inapproximability result by Vondrák [27]. However, Durr et al. [13] showed that one can bypass this inapproximability by proving approximation ratios that are functions of m, the minimum ℓ ∞ -norm of any feasible vector. Given this observation, it is possible to get results for maximizing a DR-submodular function subject to general convex set constraints, which has led to multiple works on this problem. The most recent of which is a polynomial time 1 4 (1 − m)-approximation offline algorithm due to Du [11]. However, only a sub-exponential time 1 3 √ 3 (1 − m)-approximation algorithm is known for the corresponding online problem. In this work, we present a polynomial time online algorithm matching the 1 4 (1 − m)-approximation of the state-of-the-art offline algorithm. We also present an inapproximability result showing that our online algorithm and Du's [11] offline algorithm are both optimal in a strong sense. Finally, we study the empirical performance of our algorithm and the algorithm of Du [11] (which was only theoretically studied previously), and show that they consistently outperform previously suggested algorithms on revenue maximization, location summarization and quadratic programming applications.