Limit theory of combinatorial optimization for random geometric graphs

TL;DR

This paper develops strong laws of large numbers for various graph parameters in random geometric graphs under different scaling regimes, providing a unifying theoretical framework for their asymptotic behavior.

Contribution

It introduces a general theory based on subadditivity and superadditivity to establish LLNs for multiple graph parameters and optimization functionals in random geometric graphs.

Findings

LLNs for domination, independence, clique-covering, eternal domination, and triangle packing numbers.

LLNs for minimum weight problems like TSP, spanning tree, and matchings under polynomial growth weight functions.

Unified theoretical approach applicable in thermodynamic and dense limits.

Abstract

In the random geometric graph $G(n,r_n)$, $n$ vertices are placed randomly in Euclidean $d$-space and edges are added between any pair of vertices distant at most $r_n$ from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for $G(n,r_n)$ in the thermodynamic limit with $nr_n^d =$ const., and also in the dense limit with $n r_n^d \to \infty$, $r_n \to 0$. 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 travelling salesman, spanning tree, matching, bipartite matching and bipartite travelling salesman problems, for a general class of weight functions with at most polynomial growth of order $d-\varepsilon$, under thermodynamic scaling of the distance parameter.