  Many algorithms for maximizing a monotone submodular function subject to a
knapsack constraint rely on the natural greedy heuristic. We present a novel
refined analysis of this greedy heuristic which enables us to: $(1)$ reduce the
enumeration in the tight $(1-e^{-1})$-approximation of [Sviridenko 04] from
subsets of size three to two; $(2)$ present an improved upper bound of
$0.42945$ for the classic algorithm which returns the better between a single
element and the output of the greedy heuristic.
