  We introduce the problem of maximizing approximately $k$-submodular functions
subject to size constraints. In this problem, one seeks to select $k$-disjoint
subsets of a ground set with bounded total size or individual sizes, and
maximum utility, given by a function that is "close" to being $k$-submodular.
The problem finds applications in tasks such as sensor placement, where one
wishes to install $k$ types of sensors whose measurements are noisy, and
influence maximization, where one seeks to advertise $k$ topics to users of a
social network whose level of influence is uncertain. To deal with the problem,
we first provide two natural definitions for approximately $k$-submodular
functions and establish a hierarchical relationship between them. Next, we show
that simple greedy algorithms offer approximation guarantees for different
types of size constraints. Last, we demonstrate experimentally that the greedy
algorithms are effective in sensor placement and influence maximization
problems.
