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December 2021 Limit theory of combinatorial optimization for random geometric graphs
Dieter Mitsche, Mathew D. Penrose
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Ann. Appl. Probab. 31(6): 2721-2771 (December 2021). DOI: 10.1214/20-AAP1661


In the random geometric graph G(n,rn), n vertices are placed randomly in Euclidean d-space and edges are added between any pair of vertices distant at most rn from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for G(n,rn) in the thermodynamic limit with nrnd= const., and also in the dense limit with nrnd, rn0. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the traveling salesman, spanning tree, matching, bipartite matching and bipartite traveling salesman problems, for a general class of weight functions with at most polynomial growth of order dε, under thermodynamic scaling of the distance parameter.

Funding Statement

D. Mitsche has been partially supported by grant GrHyDy ANR-20-CE40-0002, by IDEXLYON of Université de Lyon (Programme Investissements d’Avenir ANR16-IDEX-0005) and by Labex MILYON/ANR-10-LABX-0070.


We thank Joe Yukich for suggesting that we address the issue of rates of convergence. We also thank the referee for some helpful comments.


Download Citation

Dieter Mitsche. Mathew D. Penrose. "Limit theory of combinatorial optimization for random geometric graphs." Ann. Appl. Probab. 31 (6) 2721 - 2771, December 2021.


Received: 1 May 2020; Revised: 1 November 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/20-AAP1661

Primary: 05C80 , 60D05 , 60F15
Secondary: 60G55 , 90C27

Keywords: clique-covering number , dense limit , domination number , independence number , minimum-weight matching , Random geometric graph , sphere packing , subadditivity‎ , thermodynamic limit , Traveling salesman problem

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.31 • No. 6 • December 2021
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