  We extend the work of Narasimhan and Bilmes [30] for minimizing set functions
representable as a dierence between submodular functions. Similar to [30], 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 [30], and our algorithms can be used to
efficiently minimize a dierence between submodular functions under various
combinatorial constraints, a problem not previously addressed. We provide
computational bounds and a hardness result on the multiplicative
inapproximability of minimizing the dierence between submodular functions. We
show, however, that it is possible to give worst-case additive bounds by
providing a polynomial time computable lower-bound on the minima. Finally we
show how a number of machine learning problems can be modeled as minimizing the
dierence between submodular functions. We experimentally show the validity of
our algorithms by testing them on the problem of feature selection with
submodular cost features.
