How Do You Want Your Greedy: Simultaneous or Repeated?

We present SIMULTANEOUSGREEDYS, a deterministic algorithm for constrained submodular maximization. At a high level, the algorithm maintains l solutions and greedily updates them in a simultaneous fashion. SIMULTANEOUSGREEDYS achieves the tightest known which are (k + 1)2/k = k + O(1) and (1 + root k + 2)2 = k + O( approximation guarantees for both k-extendible systems and the more general k-systems, root k), respectively. We also improve the analysis of REPEATEDGREEDY, showing that it achieves an approximation root root ratio of k+O( k) for k-systems when allowed to run for O( k) iterations, an improvement in both the runtime and approximation over previous analyses. We demonstrate that both algorithms may be modified to run in nearly linear time with an arbitrarily small loss in the approximation. Both SIMULTANEOUSGREEDYS and REPEATEDGREEDY are flexible enough to incor-porate the intersection of m additional knapsack constraints, while retaining similar ap-proximation guarantees: both algorithms yield an approximation guarantee of roughly k + 2m + O(root k + m) for k-systems and SIMULTANEOUSGREEDYS enjoys an improved ap-proximation guarantee of k + 2m + O(root m) for k-extendible systems. To complement our algorithmic contributions, we prove that no algorithm making polynomially many oracle queries can achieve an approximation better than k + 1/2 - epsilon. We also present SubmodularGreedy.jl, a Julia package which implements these algorithms. Finally, we test these algorithms on real datasets.