  Consider the problem of choosing a set of actions to optimize an objective
function that is a real-valued polymatroid function subject to matroid
constraints. The greedy strategy provides an approximate solution to the
optimization problem, and it is known to satisfy some performance bounds in
terms of the total curvature. The total curvature depends on the value of
objective function on sets outside the constraint matroid. If we are given a
function defined only on the matroid, the problem still makes sense, but the
existing bounds involving the total curvature do not apply. This is puzzling:
If the optimization problem is perfectly well defined, why should the bounds no
longer apply? This motivates an alternative formulation of such bounding
techniques. The first question that comes to mind is whether it is possible to
extend a polymatroid function defined on a matroid to one on the entire power
set. This was recently shown to be negative in general. Here, we provide
necessary and sufficient conditions for the existence of an \emph{incremental}
extension of a polymatroid function defined on the uniform matroid of rank $k$
to one defined on the uniform matroid of rank $k+1$, together with an algorithm
for constructing the extension. Whenever a polymatroid objective function
defined on a matroid can be extended to the entire power set, the greedy
approximation bounds involving the total curvature of the extension apply.
However, these bounds still depend on sets outside the constraint matroid.
Motivated by this, we define a new notion of curvature called \emph{partial
curvature}, involving only sets in the matroid. We derive necessary and
sufficient conditions for an extension of the function to have a total
curvature that is equal to the partial curvature. Moreover, we prove that the
bounds in terms of the partial curvature are in general improved over the
previous ones......
