We investigate a more generalized form of submodular maximization, referred to as k-submodular maximization, with applications across social networks and machine learning domains. In this work, we propose the multilinear extension of k-submodular functions and unified Frank-Wolfe-type frameworks based on that. Our frameworks accomodate 1) monotone or non-monotone functions, and 2) various constraint types including matroid constraints, knapsack constraints, and their combinations. Notably, we attain an asymptotically optimal 1/2-approximation for monotone k-submodular maximization problems with knapsack constraints, surpassing the previous 1/3-approximation. The foundation for our analysis stems from new insights into specific linear and monotone properties pertaining to the multilinear extension.
@article{arxiv.2107.07103,
title = {On Optimal Approximations for $k$-Submodular Maximization via Multilinear Extension},
author = {Lingxiao Huang and Baoxiang Wang and Huanjian Zhou},
journal= {arXiv preprint arXiv:2107.07103},
year = {2023}
}