For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal values of various problems. However, there has not been as much success in proving central limit theorems. This paper introduces a method for establishing central limit theorems in the sparse graph setting. It works for problems that display a key property which has been variously called “endogeny,” “long-range independence” and “replica symmetry” in the literature. Examples of such problems are maximum weight matching, λ-diluted minimum matching, and optimal edge cover.
Research was supported by NSF Grant DMS RTG 1501767.
We thank Sourav Chatterjee for helpful conversations and encouragement. We also thank the anonymous referees for many helpful comments and suggestions.
"Central limit theorems for combinatorial optimization problems on sparse Erdős–Rényi graphs." Ann. Appl. Probab. 31 (4) 1687 - 1723, August 2021. https://doi.org/10.1214/20-AAP1630