The Complexity of MaxMin Length Triangulation

In 1991, Edelsbrunner and Tan gave an O(n^2) algorithm for finding the MinMax Length triangulation of a set of points in the plane. In this paper we resolve one of the open problems stated in that paper, by showing that finding a MaxMin Length triangulation is an NP-complete problem. The proof implies that (unless P=NP), there is no polynomial-time approximation algorithm that can approximate the problem within any polynomial factor.

💡 Research Summary

The paper tackles the “Maximum‑Minimum Edge Length Triangulation” (MaxMin Triangulation) problem: given a set P of n points in the plane, find a triangulation whose shortest edge is as long as possible. While the complementary “MinMax” problem (minimizing the longest edge) was solved by Edelsbrunner and Tan in 1991 with an O(n²) algorithm, the complexity of the MaxMin variant remained an open question. The authors resolve this by proving that MaxMin Triangulation is NP‑complete and, consequently, admits no polynomial‑time approximation algorithm with any polynomial‑factor guarantee unless P = NP.

The proof proceeds via a polynomial‑time reduction from planar 3‑SAT, a known NP‑complete problem. The reduction constructs a planar point set together with a set of geometric “gadgets” that encode variables, clauses, and the propagation of truth assignments. Each variable gadget can be triangulated in exactly two ways, corresponding to the true/false assignment of that variable. These choices force a specific triangulation of the “wire” segments that connect the variable gadgets to clause gadgets. A clause gadget is designed so that its local triangulation can achieve a prescribed minimum edge length L only if at least one of its three incoming wires carries the “true” configuration; otherwise the gadget forces an edge shorter than L (by a small margin ε). By setting all other edges in the construction to be longer than L, the global minimum edge length of the entire triangulation equals L if and only if the original planar 3‑SAT formula is satisfiable.

The authors formalize this intuition through a series of lemmas: (1) variable gadgets admit exactly two mutually exclusive triangulations; (2) wire segments faithfully transmit the chosen truth value without creating shorter edges; (3) clause gadgets enforce the “at least one true literal” condition; (4) the overall layout remains planar and free of unintended intersections; and (5) the size of the constructed point set is polynomial in the size of the 3‑SAT instance. Combining these lemmas yields the main theorem: MaxMin Triangulation is NP‑hard, and because a candidate triangulation can be checked in polynomial time, the problem is NP‑complete.

A further corollary follows from the ability to amplify the gap between the “yes” and “no” cases arbitrarily: for any polynomial function α(n), an α‑approximation algorithm for MaxMin Triangulation would solve planar 3‑SAT in polynomial time, implying P = NP. Hence no polynomial‑factor approximation exists unless the widely believed separation of complexity classes holds.

The paper concludes by discussing the broader implications of this hardness result. It shows that MaxMin objectives, which intuitively seem “dual” to the well‑studied MinMax objectives, can be dramatically harder. The authors suggest future work on restricted point configurations (e.g., points on a convex hull, grid points) where the problem might become tractable, as well as the development of heuristic or exact exponential‑time algorithms for practical instances. The result also invites a re‑examination of other geometric optimization problems that involve maximizing a minimum measure, as similar reductions may apply.