On the minimum spanning tree problem in imprecise set-up

TL;DR

This paper investigates the Euclidean minimum spanning tree problem with neighborhoods represented as non-crossing line segments, establishing NP-hardness and proposing approximation and parameterized algorithms for the problem.

Contribution

The paper proves NP-hardness of the problem and introduces a $2eta$-factor approximation algorithm along with a parameterized algorithm based on segment separability.

Findings

NP-hardness of the problem in general case

A $2eta$-factor approximation algorithm for the problem

A parameterized algorithm based on segment separability

Abstract

In this article, we study the Euclidean minimum spanning tree problem in an imprecise setup. The problem is known as the \emph{Minimum Spanning Tree Problem with Neighborhoods} in the literature. We study the problem where the neighborhoods are represented as non-crossing line segments. Given a set ${\cal S}$ of $n$ disjoint line segments in $I\!\!R^2$, the objective is to find a minimum spanning tree (MST) that contains exactly one end-point from each segment in $\cal S$ and the cost of the MST is minimum among $2^n$ possible MSTs. We show that finding such an MST is NP-hard in general, and propose a $2\alpha$-factor approximation algorithm for the same, where $\alpha$ is the approximation factor of the best-known approximation algorithm to compute a minimum cost Steiner tree in an undirected graph with non-negative edge weights. As an implication of our reduction, we can show that the unrestricted version of the problem (i.e., one point must be chosen from each segment such that the cost of MST is as minimum as possible) is also NP-hard. We also propose a parameterized algorithm for the problem based on the "separability" parameter defined for segments.