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.

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