The Overlap Gap Property: a Geometric Barrier to Optimizing over Random Structures

Abstract

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 possible. At the same time the formal hardness of these problems in form of say complexity-theoretic N P -hardness is lacking.In this introductory article a new approach for algorithmic intractability in random structures is described, which is based on the topological disconnectivity property of the set of pair-wise distances of near optimal solutions, called the Overlap Gap Property. The article demonstrates how this property a) emerges in most models known to exhibit an apparent algorithmic hardness b) is consistent with the hardness/tractability phase transition for many models analyzed to the day, and importantly c) allows to mathematically rigorously rule out large classes of algorithms as potential contenders, in particular the algorithms exhibiting the input stability (insensitivity).

The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model

“…It is also possible that the techniques used here can be applied to higher spin models such as Max k-XOR for k larger than 2. These models are believed to be more challenging for classical algorithms, and the performance of the QAOA may or may not face the same obstacles [10]. Can one find other problems at high qubit number and high depth where the performance of the QAOA can be established?…”

“…There may be a classical message-passing algorithm that can do better on MaxCut on large random D-regular graphs, assuming the solution space has no "overlap gap property" (see[10] for an overview). This is based on an algorithm developed for a related problem[11,12] whose success is also contingent on the assumption of no overlap gap property.…”