We analyze the performance of the greedy algorithm, and also a discrete semi-gradient based algorithm, for maximizing the sum of a suBmodular and suPermodular (BP) function (both of which are non-negative monotone non-decreasing) under two types of constraints, either a cardinality constraint or $p\geq 1$ matroid independence constraints. These problems occur naturally in several real-world applications in data science, machine learning, and artificial intelligence. The problems are ordinarily inapproximable to any factor (as we show). Using the curvature $κ_f$ of the submodular term, and introducing $κ^g$ for the supermodular term (a natural dual curvature for supermodular functions), however, both of which are computable in linear time, we show that BP maximization can be efficiently approximated by both the greedy and the semi-gradient based algorithm. The algorithms yield multiplicative guarantees of $\frac{1}{κ_f}\left[1-e^{-(1-κ^g)κ_f}\right]$ and $\frac{1-κ^g}{(1-κ^g)κ_f + p}$ for the two types of constraints respectively. For pure monotone supermodular constrained maximization, these yield $1-κ^g$ and $(1-κ^g)/p$ for the two types of constraints respectively. We also analyze the hardness of BP maximization and show that our guarantees match hardness by a constant factor and by $O(\ln(p))$ respectively. Computational experiments are also provided supporting our analysis.
Submodular function maximization has found a wealth of new applications in machine learning models during the past years. The related supermodular maximization models (submodular minimization) also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints. Hence, while there are well-developed tools for maximizing a submodular function subject to a matroid constraint, there is much less work on the corresponding supermodular maximization problems. We give a broad parameterized family of monotone functions which includes submodular …
We consider classes of objective functions of cardinality constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of $γ$ - $α$ -augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, $α$ -augmentable functions, and weighted rank functions of an independence system of bounded rank quotient - as well as additional objective functions for which the greedy algorithm yields an approximation. For this general class of functions, we show a tight bound of …
Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint
National University of Singapore（新加坡国立大学）；Nanyang Technological University（南洋理工大学）；University of Science and Technology of China（中国科学技术大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）
