  The problem of objectively choosing a string of actions to optimize an
objective function that is string submodular has been considered in [1]. There
it is shown that the greedy strategy, consisting of a string of actions that
only locally maximizes the step-wise gain in the objective function achieves at
least a (1-e^{-1})-approximation to the optimal strategy. This paper improves
this approximation by introducing additional constraints on curvatures, namely,
total backward curvature, total forward curvature, and elemental forward
curvature. We show that if the objective function has total backward curvature
\sigma, then the greedy strategy achieves at least a
\frac{1}{\sigma}(1-e^{-\sigma})-approximation of the optimal strategy. If the
objective function has total forward curvature \epsilon, then the greedy
strategy achieves at least a (1-\epsilon)-approximation of the optimal
strategy. Moreover, we consider a generalization of the diminishing-return
property by defining the elemental forward curvature. We also consider the
problem of maximizing the objective function subject to general a
string-matroid constraint. We investigate an applications of string submodular
functions with curvature constraints.
