On maximizing monotone or non-monotone k-submodular functions with the intersection of knapsack and matroid constraints

A k-submodular function is a generalization of a submodular function. The definition domain of a k-submodular function is a collection of k-disjoint subsets instead of simple subsets of ground set. In this paper, we consider the maximization of a k-submodular function with the intersection of a knapsack and m matroid constraints. When the k-submodular function is monotone, we use a special analytical method to get an approximation ratio 1m+2(1-e-(m+2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{m+2}(1-e^{-(m+2)})$$\end{document} for a nested greedy and local search algorithm. For non-monotone case, we can obtain an approximate ratio 1m+3(1-e-(m+3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{m+3}(1-e^{-(m+3)})$$\end{document}.