Central limit theorems for combinatorial optimization problems on sparse Erd\H{o}s-R\'enyi graphs

For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal value 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 which 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, $\lambda$-diluted minimum matching, and optimal edge cover.