2020/09/29 by Moran Feldman, Christopher Harshaw, Feldman, Moran +3 · 1 citation
Computer Science · #Complexity and Algorithms in Graphs #Computational Geometry and Mesh Generation #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Optimization and Control (math.OC) #Optimization and Search Problems
paper · pdf · doi:10.48550/arxiv.2009.13998
openalex publication_date 2020/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 approximation guarantees for both k-extendible systems and the more general k-systems, which are (k+1)2/k = k + O(1) and (1 + √(k+2))2 = k + O(√(k)), respectively. This is in contrast to previous algorithms, which are designed to provide tight approximation guarantees in one setting, but not both. We also improve the analysis of RepeatedGreedy, showing that it achieves an approximation 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 incorporate the intersection of m additional knapsack constraints, while retaining similar approximation guarantees: both algorithms yield an approximation guarantee of roughly k + 2m + O(√(k+m)) for k-systems and SimultaneousGreedys enjoys an improved approximation guarantee of k+2m + O(√(m)) for k-extendible systems. To complement our algorithmic contributions, we provide a hardness result which states that no algorithm making polynomially many oracle queries can achieve an approximation better than k + 1/2 + ε. We also present SubmodularGreedy.jl, a Julia package which implements these algorithms and may be downloaded at https://github.com/crharshaw/SubmodularGreedy.jl . Finally, we test the effectiveness of these algorithms on real datasets.