Maximization of k-Submodular Function with a Matroid Constraint - Sun, Yunjing; Liu, Yuezhu; Li, Min (Theory and Applications of Models of Computation, 2023)
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-ε) $$(1-ε )$$ . And we obtain that the complexity of deterministic algorithm and random algorithm is O(N|D|(p+kq)) $$O(N|D|(p+kq))$$ and O(|D|(plogNε1+kqlogNε2)logN) $$O(|D|(p\log \frac{N}{ε _1}+ k q\log \frac{N}{ε _2})\log N)$$ 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 ε,ε1,ε2 $$ε , ε _1, ε _2$$ are positive parameters with ε=maxε1,ε2 $$ε =\max \{ε _1,ε _2\}$$ .