On optimal heuristic randomized semidecision procedures, with application to proof complexity

An Atypical Survey of Typical-Case Heuristic Algorithms

Heuristic approaches often do so well that they seem to pretty much always give the right answer. How close can heuristic algorithms get to always giving the right answer, without inducing seismic complexity-theoretic consequences? This article first discusses how a series of results by Berman, Buhrman, Hartmanis, Homer, Longpré, Ogiwara, Schöening, and Watanabe, from the early 1970s through the early 1990s, explicitly or implicitly limited how well heuristic algorithms can do on NP-hard problems. In particular, many desirable levels of heuristic success cannot be obtained unless severe, highl…

Ranking-Based Black-Box Complexity

Randomized search heuristics such as evolutionary algorithms, simulated annealing, and ant colony optimization are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. While several strong runtime analysis results have appeared in the last 20 years, a powerful complexity theory for such algorithms is yet to be developed. We enrich the existing notions of black-box complexity by the additional restriction that not the actual objective values, but only the relative quality of the previously evaluated solution…

经典优化问题的量子近似复杂度

Quantum Approximation Complexity of Classical Optimization Problems

本文定义量子近似复杂度类BQ-APX、BQ-PTAS和BQ-FPTAS，在NP不等于BQP假设下证明严格层级，并展示基于分解和离散对数问题的量子优势，但MaxCut等常见问题的差距仍开放。