Randomized approximation algorithms for monotone k-submodular function maximization with constraints

Abstract

In recent years, k-submodular functions have garnered significant attention due to their natural extension of submodular functions and their practical applications, such as influence maximization and sensor placement. Influence maximization involves selecting a set of nodes in a network to maximize the spread of information, while sensor placement focuses on optimizing the locations of sensors to maximize coverage or detection efficiency. This paper first proposes two randomized algorithms aimed at improving the approximation ratio for maximizing monotone k-submodular functions under matroid constraints and individual size constraints. Under the matroid constraints, we design a randomized algorithm with an approximation ratio of \(\frac{nk}{2nk-1}\) and a complexity of \(O(rn(\text {RO}+k\text {EO}))\), where n represents the total number of elements in the ground set, k represents the number of disjoint sets in a k-submodular function, r denotes the size of the largest independent set, \(\text {RO}\) indicates the time required for the matroid’s independence oracle, and \(\text {EO}\) denotes the time required for the evaluation oracle of the k-submodular function.Meanwhile, under the individual size constraints, we achieve an approximation factor of \(\frac{nk}{3nk-2}\) with a complexity of O(knB), where n is the total count of elements in the ground set, and B is the upper bound on the total size of the k disjoint subsets, belonging to \(\mathbb {Z_{+}}\). Additionally, this paper designs two double randomized algorithms to accelerate the algorithm’s running speed while maintaining the same approximation ratio, with success probabilities of (\(1-\delta \)), where \(\delta \) is a positive parameter input by the algorithms. Under the matroid constraint, the complexity is reduced to \(O(n\log r\log \frac{r}{\delta }(\text {RO}+k\text {EO}))\). Under the individual size constraint, the complexity becomes \(O(k^{2}n\log \frac{B}{k}\log \frac{B}{\delta })\).