Deterministic approximation algorithm for submodular maximization subject to a matroid constraint

• Published in: Theoretical computer science Vol. 890; pp. 1 - 15
• Main Authors: Sun, Xin, Xu, Dachuan, Guo, Longkun, Li, Min
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 12.10.2021
• Subjects: Approximation algorithm; Deterministic algorithm; Matroid constraint; Submodular maximization; Total curvature; Total curvature; Matroid constraint; Deterministic algorithm; Approximation algorithm; Submodular maximization
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2021.08.012

•Present a deterministic approximation algorithm for submodular maximization with a curvature parameter α.•Generalize the previous deterministic 0.5008-approximation due to Buchbinder et al., achieving an identical ratio of 0.5008 when α=1.•Achieve a ratio of 1 for α=0 for which the function is actually linear that the problem is polynomial solvable.•Devise more sophisticated analysis on the inequalities about monotonicity with curvature via mathematical tools such as infinite series. In this paper, we study the generalized submodular maximization problem with a non-negative monotone submodular set function as the objective function and subject to a matroid constraint. The problem is generalized through the curvature parameter α∈[0,1] which measures how far a set function deviates from linearity to submodularity. We propose a deterministic approximation algorithm which uses the approximation algorithm proposed by Buchbinder et al. [2] as a building block and inherits the approximation guarantee for α=1. For general value of the curvature parameter α∈[0,1], we present an approximation algorithm with a factor of 1+hα(y)+Δ⋅[3+α−(2+α)y−(1+α)hα(y)]2+α+(1+α)(1−y), where y∈[0,1] is a predefined parameter for tuning the ratio. In particular, when α=1 we obtain a ratio 0.5008 when setting y=0.9, coinciding with the renowned state-of-art approximate ratio; when α=0 that the object is a linear function, the approximation factor equals one and our algorithm is indeed an exact algorithm that always produces optimum solutions.