When Weak Submodularity Meets Matroids: How Does Standard Greedy Behave?

Weakly submodular maximization finds numerous applications in machine learning, data mining and statistics. In particular, under a cardinality constraint, the standard greedy algorithm is known to attain the tight $(1-e^{-γ})$-approximation guarantee, where $γ$ denotes the submodularity ratio. However, when considering general matroid constraints, how well standard greedy can perform has remained a longstanding open question. In this paper, we resolve this question by studying the worst-case approximation guarantee $R_{\mathrm{gr}}(γ,r)$ of standard forward greedy, where $r$ is the rank of matroid. Specifically, our first contribution is the following refined lower bound \begin{equation*} R_{gr}(γ,r)\ge \max\left{ \fracγ{r}, \frac{γ^2} {B_γ+\sqrt{γ(r-1)}}, \frac{γ^{r+1}} {2-γ+γ^{r+1}}, \frac{γ^2} {14\left(B_γ+\sqrt{γ(1-γ)(r-1)}\right)} \right}, \end{equation*} where $B_γ\triangleq2-γ+γ^2$. This lower bound not only strictly improves upon the previous guarantee of $0.4γ^2/(1+\sqrt{γr})$, but also can recover the classical $1/2$-approximation guarantee at $γ=1$. Subsequently, a corresponding upper bound is established for $R_{\mathrm{gr}}(γ,r)$, namely, \begin{equation}\label{eq_upper_abstract} R_{gr}(γ,r) \le\min\left{\frac{1}{2},20\max\left{\fracγ{r}, \frac{γ^2}{B_γ+ \sqrt{γ(1-γ)(r-1)}}\right}\right}. \end{equation} For every fixed $γ\in(0,1)$, this equation effectively sharpens the recent $O_γ(1/\log(n-1))$ upper bound to $O_γ(r^{-1/2})$, where $n$ is the size of ground set. Putting all these results together, we eventually can obtain

$$R_{gr}(γ,r) = Θ\left( \max\left\{\fracγ{r}, \frac{γ^2} {1+\sqrt{γ(1-γ)r}}\right\} \right).$$