Submodular + Supermodular function maximization with knapsack constraint

Abstract

We investigate a class of non-submodular function optimization problems, specifically maximizing the sum of a normalized monotone submodular function $f$ and a normalized monotone supermodular function $g$ under a knapsack constraint. By utilizing the total curvature ${\kappa }_f$ of $f$ and the supermodular curvature ${\kappa }^g$ of $g$, we demonstrate that this problem can achieve a near-optimal solution through three approaches: a greedy algorithm, an iterated submodular+modular procedure and a sandwich method. In particular, we prove that both the greedy algorithm and the iterated submodular+modular procedure provide an approximation guarantee of $\frac{1}{{\kappa }_f}(1-e^{-(1-{\kappa }^g){\kappa }_f})$, while the sandwich method achieves a $(1-{\kappa }^g)(1-\frac{{\kappa }_f}{e})$-approximation ratio. All proposed algorithms run in polynomial time, and parameters such as ${\kappa }_f$ and ${\kappa }^g$ can be computed efficiently in linear time. Additionally, all three algorithms yield a $(1-{\kappa }^g)$-approximation performance for knapsack-constrained monotone supermodular function maximization. Finally, we empirically test our first two algorithms on a constructed application. Although both algorithms have the same theoretical guarantee, their practical behavior differs significantly, leading to distinct solutions.