  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.
