Single-Source Dilation-Bounded Minimum Spanning Trees

TL;DR

This paper studies the problem of constructing a minimum total edge length geometric network from a single source point that guarantees paths within a specified dilation factor, proving NP-hardness and providing an approximation algorithm.

Contribution

It introduces the problem of single-source dilation-bounded minimum spanning trees and offers the first approximation algorithm for this NP-hard problem.

Findings

Proves the problem is NP-hard.

Develops an approximation algorithm.

Provides bounds on the approximation quality.

Abstract

Given a set $S$ of points in the plane, a geometric network for $S$ is a graph $G$ with vertex set $S$ and straight edges. We consider a broadcasting situation, where one point $r \in S$ is a designated source. Given a dilation factor $\delta$, we ask for a geometric network $G$ such that for every point $v \in S$ there is a path from $r$ to $v$ in $G$ of length at most $\delta|rv|$, and such that the total edge length is minimized. We show that finding such a network of minimum total edge length is NP-hard, and give an approximation algorithm.