On maximizing a monotone k-submodular function subject to a matroid constraint

A k-submodular function is an extension of a submodular function in that its input is given by k disjoint subsets instead of a single subset. For unconstrained nonnegative k-submodular maximization, Ward and Zivny proposed a constant-factor approximation algorithm, which was improved by the recent work of Iwata, Tanigawa and Yoshida presenting a 1/2-approximation algorithm. Iwata et al. also provided a k/(2k-1)-approximation algorithm for nonnegative monotone k-submodular maximization and proved that its approximation ratio is asymptotically tight. More recently, Ohsaka and Yoshida proposed constant-factor algorithms for nonnegative monotone k-submodular maximization with several size constraints. However, while submodular maximization with various constraints has been extensively studied, no approximation algorithm has been developed for constrained k-submodular maximization, except for the case of size constraints. In this paper, we prove that a greedy algorithm outputs a 1/2-approximate solution for nonnegative monotone k-submodular maximization with a matroid constraint. The algorithm runs in O(M vertical bar E vertical bar(IO kEO)) time, where M is the size of a maximal optimal solution, vertical bar E vertical bar is the size of the ground set, and IO, EO represent the time for the independence oracle of the matroid and the evaluation oracle of the k-submodular function, respectively. (C) 2017 Elsevier B.V. All rights reserved.