Minimum Weight Euclidean t-spanner is NP-Hard

TL;DR

This paper proves that finding a minimum weight Euclidean t-spanner for a set of points in the plane is NP-hard for any constant t greater than 1, regardless of planarity constraints.

Contribution

It establishes the NP-hardness of the Euclidean t-spanner weight optimization problem for all t > 1, filling a gap in computational geometry complexity theory.

Findings

NP-hardness for t > 1

Applies to both planar and non-planar spanners

Complexity holds for any fixed t > 1

Abstract

Given a set P of points in the plane, an Euclidean t-spanner for P is a geometric graph that preserves the Euclidean distances between every pair of points in P up to a constant factor t. The weight of a geometric graph refers to the total length of its edges. In this paper we show that the problem of deciding whether there exists an Euclidean t-spanner, for a given set of points in the plane, of weight at most w is NP-hard for every real constant t > 1, both whether planarity of the t-spanner is required or not.