New approximations for monotone submodular maximization with knapsack constraint

Given a monotone submodular set function with a knapsack constraint, its maximization problem has two types of approximation algorithms with running time \(O(n^2)\) and \(O(n^5)\), respectively. With running time \(O(n^5)\), the best performance ratio is \(1-1/e\). With running time \(O(n^2)\), the well-known performance ratio is \((1-1/e)/2\) and an improved one is claimed to be \((1-1/e^2)/2\) recently. In this paper, we design an algorithm with running \(O(n^2)\) and performance ratio \(1-1/e^{2/3}\), and an algorithm with running time \(O(n^3)\) and performance ratio 1/2.