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 non-additive objective functions, and a resolution of the worst-case approximation guarantee of local search in p-extendible systems.
@InProceedings{chatziafratis_et_al:LIPIcs.ESA.2017.26, author = {Chatziafratis, Vaggos and Roughgarden, Tim and Vondrak, Jan}, title = {{Stability and Recovery for Independence Systems}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {26:1--26:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.26}, URN = {urn:nbn:de:0030-drops-78423}, doi = {10.4230/LIPIcs.ESA.2017.26}, annote = {Keywords: Submodular, approximation, stability, Local Search, Greedy, p-systems} }
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