  We consider the problem of maximizing a fractionally subadditive function
under a knapsack constraint that grows over time. An incremental solution to
this problem is given by an order in which to include the elements of the
ground set, and the competitive ratio of an incremental solution is defined by
the worst ratio over all capacities relative to an optimum solution of the
corresponding capacity. We present an algorithm that finds an incremental
solution of competitive ratio at most $\max\{3.293\sqrt{M},2M\}$, under the
assumption that the values of singleton sets are in the range $[1,M]$, and we
give a lower bound of $\max\{2.618,M\}$ on the attainable competitive ratio. In
addition, we establish that our framework captures potential-based flows
between two vertices, and we give a lower bound of $\max\{2,M\}$ and an upper
bound of $2M$ for the incremental maximization of classical flows with
capacities in $[1,M]$ which is tight for the unit capacity case.
