Maximization of k-Submodular Function with a Matroid Constraint

• Published in: Theory and Applications of Models of Computation Vol. 13571; pp. 1 - 10
• Main Authors: Sun, Yunjing, Liu, Yuezhu, Li, Min
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2023; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Deterministic algorithm; Matroid constraint; Randomized algorithm; submodular function
• ISBN: 3031203496; 9783031203497
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-031-20350-3_1

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) $$(1-\varepsilon )$$ . 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) $$O(|D|(p\log \frac{N}{\varepsilon _1}+ k q\log \frac{N}{\varepsilon _2})\log N)$$ 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 ε,ε1,ε2 $$\varepsilon , \varepsilon _1, \varepsilon _2$$ are positive parameters with ε=maxε1,ε2 $$\varepsilon =\max \{\varepsilon _1,\varepsilon _2\}$$ .