Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-complete. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.

💡 Research Summary

The paper studies the computational complexity of determining the minimum number of diagonal flips required to transform one triangulation of a simple (non‑convex) polygon into another. A flip removes an interior diagonal of a triangulation and inserts the other diagonal of the quadrilateral formed by the two adjacent triangles, preserving a valid triangulation. The authors formalize the decision version of the problem: given a simple polygon P, two triangulations T₁ and T₂ of P, and an integer k, decide whether the flip distance between T₁ and T₂ is at most k.

First, they show that the problem belongs to NP. A certificate consists of a sequence of flips; each flip can be verified in polynomial time by checking that the removed diagonal is present, that the quadrilateral formed by its two incident triangles is convex, and that the inserted diagonal yields a valid triangulation. The final triangulation can then be compared with T₂, all in polynomial time.

The core contribution is an NP‑hardness proof via a polynomial‑time reduction from Planar Monotone Rectilinear 3‑SAT, a known NP‑complete variant of SAT where variables and clauses can be placed on a grid without edge crossings and each clause is either all positive or all negative. The reduction constructs a simple polygon whose interior contains a collection of “gadgets” that simulate variables, clauses, and wires connecting them.

Variable gadget. Each variable is represented by a chain of small quadrilateral “switches.” Flipping a switch toggles the variable’s truth assignment. The gadget is designed so that committing to a truth value requires a fixed number C₁ of flips, independent of the rest of the construction.

Clause gadget. For each clause, a “clause region” is attached to the switches of the literals that appear in the clause. The region is built so that it can be kept in a low‑cost configuration only if at least one of its incident variable switches is set to the satisfying polarity. If all incident switches are set opposite to the clause’s requirement, the clause region forces additional flips (cost C₂ > C₁) to be resolved.

Wires. Thin corridors of quadrilaterals act as wires that transmit the effect of a variable flip to all clause gadgets that contain that variable. The geometry guarantees that a flip in a variable switch propagates along the wire without creating illegal triangulations.

All gadgets are assembled inside a single simple polygon. The total number of vertices of the polygon is linear in the size of the SAT instance, ensuring a polynomial‑size reduction. The authors define a threshold K = (number of variables)·C₁ + (number of clauses)·C₂. They prove two directions:

1. If the SAT instance is satisfiable, there exists a flip sequence that sets each variable gadget according to a satisfying assignment (cost C₁ per variable) and then resolves each clause gadget in its low‑cost mode (cost C₂ per clause). The total number of flips is exactly K.

2. If the SAT instance is unsatisfiable, any assignment of the variable gadgets leaves at least one clause gadget in an unsatisfied state, which forces extra flips beyond C₂ for that clause. Consequently any flip sequence requires more than K flips.

Thus the decision problem “flip distance ≤ K?” is equivalent to the original SAT instance, establishing NP‑hardness. Combined with the NP membership, the flip‑distance problem for simple polygons is NP‑complete.

The paper situates this result within the broader literature. For convex polygons, the flip graph is isomorphic to the associahedron, and the flip distance coincides with the rotation distance between binary trees—a long‑standing open problem with no known polynomial‑time algorithm. For arbitrary planar point sets, recent work has shown APX‑hardness of the flip‑distance problem, indicating that even approximation is difficult. The present work shows that the hardness already appears when the point set is the vertex set of a single simple polygon, highlighting the dramatic impact of non‑convexity on the combinatorial structure of the flip graph.

Finally, the authors discuss future directions: (i) designing approximation or fixed‑parameter algorithms for simple‑polygon flip distance, (ii) identifying subclasses of simple polygons (e.g., y‑monotone, star‑shaped) where the problem might become tractable, and (iii) studying structural properties of the flip graph such as diameter, connectivity, and the existence of efficient routing schemes. The NP‑completeness result sets a clear boundary for exact algorithms and motivates the search for heuristic or parameterized approaches in computational geometry and related fields.