Approximation Guarantees for Deterministic Maximization of Submodular Function with a Matroid Constraint

Abstract

In the paper, we propose a deterministic approximation algorithm for maximizing a generalized monotone submodular function subject to a matroid constraint. The function is generalized through a curvature parameter $c\in [0,1]$, and essentially reduces to a submodular function when $c=1$. Our algorithm employs the deterministic approximation devised by Buchbinder et al. [3] for the $c=1$ case of the problem as a building block, and eventually attains an approximation ratio of $\frac{1+g_c(x)+\Delta \cdot \left[ 3+c-(2+c)x-(1+c)g_c(x)\right] }{2+c+(1+c)(1-x)}$ for the curvature parameter $c\in [0,1]$ and for a calibrating parameter that is any $x\in [0,1]$. For $c=1$, the ratio attains 0.5008 by setting $x=0.9$, coinciding with the renowned performance guarantee of the problem. Moreover, when the submodular set function degenerates to a linear function, our generalized algorithm always produces optimum solutions and thus achieves an approximation ratio 1.