The predominant measure for the performance of an algorithm is its worst-case approximation guarantee. While worst-case approximations give desirable robustness guarantees, they can differ significantly from the performance of an algorithm in practice. For the problem of monotone submodular maximization under a cardinality constraint, the greedy algorithm is known to obtain a 1-1/e approximation guarantee, which is optimal for a polynomial-time algorithm. However, very little is known about the approximation achieved by greedy and other submodular maximization algorithms on real instances.

We develop an algorithm that gives an instance-specific approximation for any solution of an instance of monotone submodular maximization under a cardinality constraint. This algorithm uses a novel dual approach to submodular maximization. In particular, it relies on the construction of a lower bound to the dual objective that can also be exactly minimized. We use this algorithm to show that on a wide variety of real-world datasets and objectives, greedy and other algorithms find solutions that approximate the optimal solution significantly better than the 1-1/e ~ 0.63 worst-case approximation guarantee, often exceeding 0.9.