Using random spanning trees in survivable networks design

TL;DR

This paper analyzes a process of joining multiple random spanning trees to create highly connected graphs, deriving formulas for their expected edges and designing an efficient algorithm for survivable network design with near-optimal solutions.

Contribution

It introduces a novel method of combining random spanning trees to generate $k$-edge connected graphs and provides an algorithm with provable approximation guarantees.

Findings

Expected number of edges in the generated graph is derived.

An upper bound on the concentration of the number of edges is established.

An $O(kn ext{log}n)$ algorithm approximates the minimal edge count within a factor less than 2.

Abstract

We investigate a process of joining $k$ random spanning trees on a fixed clique $K_n$. The joined trees may not be disjoint and multiple edges are replaced by one simple edge. This process produces a simple graph $G$ on $n$~vertices with an edge set, which is a union of edge sets of the joined trees. We study a random variable $S_{k}$ of the number of edges in the generated graph $G$. The exact formula is derived for the expected value of the random variable $S_{k}$. In addition, an upper bound on the concentration coefficient of the random variable $S_{k}$ is provided. We use results of our analysis to design an algorithm to generate $k$-edge connected graphs for arbitrarily large values of $k \geq 2$. The designed algorithm solves a particular case of the Survivable Network Design Problem, where the cost of each edge is $c_{e} = 1$ and the connectivity requirement for each pair of vertices $u, v \in V(G)$ is $k$.The proposed algorithm is within a factor strictly less than $2$ of the optimal value (i.e., the number of edges in the generated graph) and its running time is $O(kn\log{n})$.