Scaling Submodular Maximization via Pruned Submodularity Graphs

We propose a new randomized pruning method (called “submodular sparsification (SS)”) to reduce the cost of submodular maximization. The pruning is applied via a “submodularity graph” over the n ground elements, where each directed edge is associated with a pairwise dependency defined by the submodular function. In each step, SS prunes a 1 1/ p c (for c > 1) fraction of the nodes using weights on edges computed based on only a small number (O(log n)) of randomly sampled nodes. The algorithm requires logp c n steps with a small and highly parallelizable per-step computation. An accuracy-speed tradeoff parameter c, set e.g. as c = 8, leads to a fast shrink rate p 2/4 and small iteration complexity log