Maximizing approximately k-submodular functions

TL;DR

This paper studies the problem of maximizing functions that are nearly $k$-submodular under size constraints, proposing greedy algorithms with proven approximation guarantees and demonstrating their effectiveness in sensor placement and influence maximization tasks.

Contribution

It introduces formal definitions for approximately $k$-submodular functions, analyzes greedy algorithms for this setting, and validates their practical effectiveness through experiments.

Findings

Greedy algorithms achieve approximation guarantees for approximately $k$-submodular maximization.

Hierarchical relationships between different definitions of approximate $k$-submodularity are established.

Experimental results confirm the algorithms' effectiveness in sensor placement and influence maximization.

Abstract

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.