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We investigate a more generalized form of submodular maximization, referredto as $k$-submodular maximization, with applications across social networks andmachine 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) variousconstraint types including matroid constraints, knapsack constraints, and theircombinations. Notably, we attain an asymptotically optimal $1/2$-approximationfor monotone $k$-submodular maximization problems with knapsack constraints,surpassing the previous $1/3$-approximation. The foundation for our analysisstems from new insights into specific linear and monotone properties pertainingto the multilinear extension.

On Optimal Approximations for $k$-Submodular Maximization via Multilinear Extension