A $k$-submodular function is a function that given $k$ disjoint subsets outputs a value that is submodular in every orthant. In this paper, we provide a new framework for $k$-submodular maximization problems, by relaxing the optimization to the continuous space with the multilinear extension of $k$-submodular functions and a variant of pipage rounding that recovers the discrete solution. The multilinear extension introduces new techniques to analyze and optimize $k$-submodular functions. When the function is monotone, we propose almost $\frac{1}{2}$-approximation algorithms for unconstrained maximization and maximization under total size and knapsack constraints. For unconstrained monotone and non-monotone maximization, we propose an algorithm that is almost as good as any combinatorial algorithm based on Iwata, Tanigawa, and Yoshida's meta-framework ($\frac{k}{2k-1}$-approximation for the monotone case and $\frac{k^2+1}{2k^2+1}$-approximation for the non-monotone case).

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