The overlap gap property: A topological barrier to optimizing over random structures
Author(s)
Gamarnik, David
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The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures, finding optimal solutions by means of fast algorithms is not known and often is believed not to be possible. At the same time, the formal hardness of these problems in the form of the complexity-theoretic
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-hardness is lacking. A new approach for algorithmic intractability in random structures is described in this article, which is based on the topological disconnectivity property of the set of pairwise distances of near-optimal solutions, called the Overlap Gap Property. The article demonstrates how this property 1) emerges in most models known to exhibit an apparent algorithmic hardness; 2) is consistent with the hardness/tractability phase transition for many models analyzed to the day; and, importantly, 3) allows to mathematically rigorously rule out a large class of algorithms as potential contenders, specifically the algorithms that exhibit the input stability (insensitivity).
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Date issued
2021Department
Massachusetts Institute of Technology. Operations Research Center; Sloan School of ManagementJournal
Proceedings of the National Academy of Sciences of the United States of America
Publisher
Proceedings of the National Academy of Sciences
Citation
Gamarnik, David. 2021. "The overlap gap property: A topological barrier to optimizing over random structures." Proceedings of the National Academy of Sciences of the United States of America, 118 (41).
Version: Final published version