Streaming Algorithms for Constrained Submodular Maximization

Abstract

Due to the pervasive "diminishing returns" property appeared in data-intensive applications, submodular maximization problems have aroused great attention from both the machine learning community and the computation theory community. During the last decades, a lot of algorithms have been proposed for submodular maximization subject to various constraints [4, 6, 8], and these algorithms can be used in numerous applications including sensor placement [9], clustering [5], network design [13], and so on. The existing algorithms for submodular maximization can be roughly classified into offline algorithms and streaming algorithms; the former assume full access to the whole dataset at any time (e.g.,[4, 10]), while the latter only require an amount of space which is nearly linear in the maximum size of a feasible solution (e.g., [1, 7]). Apparently, streaming algorithms are more useful in big data applications, as the whole data set is usually too large to be fit into memory in practice. However, compared to the offline algorithms, the existing streaming algorithms for submodular maximization generally have weaker capabilities in that they handle more limited problem constraints or achieve weaker performance bounds, due to the more stringent requirements under the streaming setting. Another classification of the existing algorithms is that they concentrate on either monotone or non-monotone submodular functions. As monotone submodular function is a special case of non-monotone submodular function, we will concentrate on non-monotone submodular maximization in this paper.