Two genres of heuristics that are frequently reported to perform much better on "real-world" instances than in the worst case are greedy algorithms and local search algorithms. In this paper, we systematically study these two types of algorithms for the problem of maximizing a monotone submodular set function subject to downward-closed feasibility constraints. We consider perturbation-stable instances, in the sense of Bilu and Linial [11] , and precisely identify the stability threshold beyond which these algorithms are guaranteed to recover the optimal solution. Byproducts of our work include the first definition of perturbation-stability for nonadditive objective functions, and a resolution of the worst-case approximation guarantee of local search in p-extendible systems.
REF show that the greedy algorithm recovers the unique optimal solution for 2-stable instances.
Stability and Recovery for Independence Systems
ArXiv
ArXiv
Mathematics Computer Science