Maximizing a $ k $-submodular function with $ p $-system constraints

Abstract

Maximizing a k-submodular function has received widespread attention in recent years. For k-submodular maximization (k-SMM) with psystem constraints, we propose two algorithms. We first propose a greedy algorithm. Through this algorithm, we can obtain an approximation result of 1/(1 + p) if the function is monotone, and a 1/(2 + p)-approximation ratio in the non-monotone case with O(n 2 (IO + kEO)) time complexity. The second algorithm is a threshold descent algorithm. We obtain a 1 (1+ϵ)(1+p)+ϵ 2approximation ratio in the monotone case and a 1 (1+ϵ)(2+p)+ϵ 2 -approximation ratio in the non-monotone case under O( n(IO+kEO) ϵ log n ϵ ) time complexity. Although the approximation ratio is slightly reduced, the time complexity is also greatly reduced. 2020 Mathematics Subject Classification. 90C27.