Maximizing approximately k-submodular functions
This work addresses optimization challenges in domains like sensor placement and social networks, but it is incremental as it extends existing submodular function concepts to approximate settings.
The paper tackles the problem of maximizing approximately k-submodular functions under size constraints, with applications in sensor placement and influence maximization, by showing that greedy algorithms provide approximation guarantees and are effective in experiments.
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.