Fixed-Parameter Tractable Submodular Maximization over a Matroid

Abstract: In this paper, we design fixed-parameter tractable (FPT) algorithms for (non-monotone) submodular maximization subject to a matroid constraint, where the matroid rank $r$ is treated as a fixed parameter that is independent of the total number of elements $n$. We provide two FPT algorithms: one for the offline setting and another for the random-order streaming setting. Our streaming algorithm achieves a $\frac{1}{2}-\varepsilon$ approximation using $\widetilde{O}\left(\frac{r}{\textrm{poly}(\varepsilon)}\right)$ memory, while our offline algorithm obtains a $1-\frac{1}{e}-\varepsilon$ approximation with $n\cdot 2^{\widetilde{O}\left(\frac{r}{\textrm{poly}(\varepsilon)}\right)}$ runtime and $\widetilde{O}\left(\frac{r}{\textrm{poly}(\varepsilon)}\right)$ memory. Both approximation factors are near-optimal in their respective settings, given existing hardness results. In particular, our offline algorithm demonstrates that--unlike in the polynomial-time regime--there is essentially no separation between monotone and non-monotone submodular maximization under a matroid constraint in the FPT framework.