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August 2021 Central limit theorems for combinatorial optimization problems on sparse Erdős–Rényi graphs
Sky Cao
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Ann. Appl. Probab. 31(4): 1687-1723 (August 2021). DOI: 10.1214/20-AAP1630

Abstract

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.

Funding Statement

Research was supported by NSF Grant DMS RTG 1501767.

Acknowledgments

We thank Sourav Chatterjee for helpful conversations and encouragement. We also thank the anonymous referees for many helpful comments and suggestions.

Citation

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Sky Cao. "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

Information

Received: 1 June 2019; Revised: 1 July 2020; Published: August 2021
First available in Project Euclid: 15 September 2021

Digital Object Identifier: 10.1214/20-AAP1630

Subjects:
Primary: 60F05
Secondary: 82B44 , 90C27

Keywords: central limit theorem , Combinatorial optimization , endogeny , Erdős–Rényi graph , generalized perturbative approach , long-range independence , maximum weight matching , Minimum matching , optimal edge cover , replica symmetry , Stein’s method

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 4 • August 2021
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