A $(k+3)\mathrm{/}2$-approximation algorithm for monotone submodular maximization over a $k$-exchange system

We consider the problem of maximizing a monotone submodular function in a $k$-exchange system. These systems, introduced by Feldman et al., generalize the matroid k-parity problem in a wide class of matroids and capture many other combinatorial optimization problems. Feldman et al. show that a simple non-oblivious local search algorithm attains a $(k+1)\mathrm{/}2$ approximation ratio for the problem of linear maximization in a $k$-exchange system. Here, we extend this approach to the case of monotone submodular objective functions. We give a deterministic, non-oblivious local search algorithm that attains an approximation ratio of $(k+3)\mathrm{/}2$ for the problem of maximizing a monotone submodular function in a $k$-exchange system.