On Minimum Spanning Trees for Random Euclidean Bipartite Graphs

TL;DR

This paper analyzes the minimum spanning tree problem on random bipartite graphs with Euclidean edge weights, revealing growth in maximum degree and establishing convergence of normalized total costs, extending classical results to bipartite cases.

Contribution

It extends classical MST results to bipartite graphs with Euclidean weights, showing degree growth and cost convergence in the large n limit.

Findings

Maximum vertex degree grows logarithmically with n

Normalized total edge costs converge to a constant for p<d

Results extend classical non-bipartite MST theory

Abstract

We consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$-th power of their Euclidean distance, with $p>0$. In the large $n$ limit with $n_R/n \to \alpha_R$ and $0<\alpha_R<1$, we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p<d$, we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.