We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual off-line setting, we present SampleGreedy, which obtains a [Formula: see text]-approximation for maximizing a submodular function subject to a p-extendible system using [Formula: see text] evaluation and feasibility queries, where k is the size of the largest feasible set. The approximation ratio improves to p + 1 and p for monotone submodular and linear objectives, respectively. In the streaming setting, we present Sample-Streaming, which obtains a [Formula: see text]-approximation for maximizing a submodular function subject to a p-matchoid using O(k) memory and [Formula: see text] evaluation and feasibility queries per element, and m is the number of matroids defining the p-matchoid. The approximation ratio improves to 4p for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks.
Monotone Submodular Maximization over a Matroid via Non-Oblivious Local Search
