Multilinear extension of k-submodular functions

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 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 ( k 2k−1 -approximation for the monotone case and k 2 +1 2k+1 -approximation for the non-monotone case).

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