  In this paper, we introduce a novel technique for constrained submodular
maximization, inspired by barrier functions in continuous optimization. This
connection not only improves the running time for constrained submodular
maximization but also provides the state of the art guarantee. More precisely,
for maximizing a monotone submodular function subject to the combination of a
$k$-matchoid and $\ell$-knapsack constraint (for $\ell\leq k$), we propose a
potential function that can be approximately minimized. Once we minimize the
potential function up to an $\epsilon$ error it is guaranteed that we have
found a feasible set with a $2(k+1+\epsilon)$-approximation factor which can
indeed be further improved to $(k+1+\epsilon)$ by an enumeration technique. We
extensively evaluate the performance of our proposed algorithm over several
real-world applications, including a movie recommendation system, summarization
tasks for YouTube videos, Twitter feeds and Yelp business locations, and a set
cover problem.
