Maximization of k-Submodular Function with a Matroid Constraint

A k-submodular function is a promotion of a submodular function, whose domain is composed of k disjoint subsets rather than a single subset. In this paper, we give a deterministic algorithm for the non-monotone k-submodular function maximization problem subject to a matroid constraint with approximation factor 1/3. Based on this result, we give a randomized 1/3-approximation algorithm for the problem with faster running time, but the probability of success is $$(1-\varepsilon )$$ . And we obtain that the complexity of deterministic algorithm and random algorithm is $$O(N|D|(p+kq))$$ and $$O(|D|(p\log \frac{N}{\varepsilon _1}+ k q\log \frac{N}{\varepsilon _2})\log N)$$ respectively, where D is the ground set of the matroid constraint with rank N, p is times of oracle to calculate whether a set is an independent set in this matroid, q is the times of oracle to calculate a value of the k-submodular function, and $$\varepsilon , \varepsilon _1, \varepsilon _2$$ are positive parameters with $$\varepsilon =\max \{\varepsilon _1,\varepsilon _2\}$$ .