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We extend the work of Narasimhan and Bilmes  for minimizing set functions representable as a difference between submodular functions. Similar to , our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than , and our algorithms can be used to efficiently minimize a difference between submodular functions under various combinatorial constraints, aarXiv:1207.0560v4 fatcat:nhrbsck73jezbkmhqoqgzuijq4