Minimum Weight Euclidean t-spanner is NP-Hard

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.

💡 Research Summary

The paper addresses the decision problem of whether a given set P of points in the Euclidean plane admits a t‑spanner whose total edge length does not exceed a prescribed bound w, for any fixed real constant t > 1. A t‑spanner is a geometric graph G = (P, E) such that for every pair of points u, v ∈ P the shortest‑path distance d_G(u, v) is at most t times the Euclidean distance ‖u − v‖. The weight of G is defined as the sum of the Euclidean lengths of all edges in E. The authors prove that this decision problem is NP‑hard, and they show that the hardness persists even when the spanner is required to be planar (i.e., its edges may not cross).

The proof proceeds by a polynomial‑time reduction from the classic NP‑complete PARTITION problem. An instance of PARTITION consists of a multiset A = {a₁,…,aₙ} of positive integers and the target value B = ½ ∑ aᵢ. The reduction constructs a point set P that encodes each integer aᵢ as a geometric “gadget”. Each gadget consists of a vertical segment of length proportional to aᵢ together with a short horizontal segment, forming a small L‑shaped structure. The gadget admits two alternative ways of being connected inside any feasible spanner:

Selection path – the spanner follows the vertical segment directly, thereby contributing exactly aᵢ to the total weight.
Non‑selection path – the spanner bypasses the vertical segment via a detour of negligible additional length (ε).

All gadgets are placed along a large rectangular frame whose perimeter has a fixed length w₀. The overall weight budget is set to w = w₀ + B. If the PARTITION instance has a subset S ⊆ A whose sum equals B, then a spanner can be built by choosing the selection path for precisely those gadgets corresponding to elements of S and the non‑selection path for the remaining gadgets. The total weight contributed by the selected gadgets is exactly B, so the overall weight meets the bound w. Conversely, if no such subset exists, any combination of selection and non‑selection paths will either fall short of B (making the spanner too light, which is impossible because the frame already forces weight w₀) or exceed B, thereby violating the weight limit. Hence a feasible spanner of weight ≤ w exists if and only if the original PARTITION instance is solvable.

To extend the result to planar spanners, the authors carefully arrange the gadgets on a grid so that any edge required by the construction can be drawn without crossing another. They introduce “cross‑prevention lines”—short auxiliary edges that must be present in any valid spanner—to enforce planarity without affecting the weight analysis. By spacing gadgets sufficiently far apart, the selection and non‑selection paths of different gadgets remain disjoint, guaranteeing that the resulting spanner is planar.

The reduction runs in time polynomial in n, and the size of the constructed point set is linear in the size of the PARTITION instance. Consequently, the decision problem “Does there exist a Euclidean t‑spanner of weight at most w?” is NP‑hard for every fixed t > 1, both in the unrestricted and planar settings.

The paper concludes by discussing the implications of this hardness result. Since finding the exact minimum‑weight t‑spanner is computationally intractable, research should focus on approximation algorithms, fixed‑parameter tractable approaches, or special cases (e.g., points on a grid, bounded t, or bounded degree). The authors also note that their reduction technique could be adapted to prove hardness for related geometric network design problems, such as minimum‑weight Steiner t‑spanners or fault‑tolerant spanners.