Augmenting a Geometric Matching is $NP$-complete

Given $2n$ points in the plane, it is well-known that there always exists a perfect straight-line non-crossing matching. We show that it is $NP$-complete to decide if a partial matching can be augmented to a perfect one, via a reduction from 1-in-3-SAT. This result also holds for bichromatic matchings.

💡 Research Summary

The paper investigates the computational complexity of extending a given partial straight‑line non‑crossing matching on a set of 2n planar points to a perfect matching. While it is a classic result that any set of 2n points in general position admits a perfect non‑crossing straight‑line matching, the authors ask a more nuanced decision problem: given some edges already fixed (a partial matching), can the remaining points be paired by additional straight‑line segments so that the final set of n edges is still non‑crossing?

To answer this, the authors prove that the problem is NP‑complete. The hardness proof proceeds via a polynomial‑time reduction from the well‑known NP‑complete problem 1‑in‑3‑SAT, where each clause must have exactly one true literal. The reduction constructs a geometric instance consisting of two families of gadgets: variable gadgets and clause gadgets.

A variable gadget consists of a small configuration of points that admits exactly two possible non‑crossing matchings, representing the truth assignment “true” or “false”. Each of these two matchings exposes a distinct “port” that can be connected to clause gadgets. By selecting one of the two matchings, the gadget “activates” the corresponding port while the other port remains unused.

A clause gadget is more intricate. It contains three input ports, each intended to be linked to a different variable gadget that appears in the clause. Inside the clause gadget a set of fixed points and “blocker” points are arranged so that the gadget can be completed to a perfect matching only if exactly one of its three ports is already occupied by a line from a variable gadget. If zero or two (or more) ports are active, any attempt to finish the gadget inevitably creates a crossing, violating the non‑crossing requirement. This geometric enforcement mirrors the 1‑in‑3 condition.

All gadgets are placed on the integer lattice in a planar fashion that guarantees no unintended crossings between edges belonging to different gadgets. The total number of points is linear in the size of the SAT formula, and the construction can be carried out in polynomial time.

The correctness of the reduction is established in two directions. (1) If the original 1‑in‑3‑SAT instance is satisfiable, assign each variable gadget the matching corresponding to its truth value. Because each clause has exactly one true literal, exactly one port of each clause gadget is activated, allowing the clause gadget to be completed without crossings. Consequently the whole partial matching can be extended to a perfect non‑crossing matching. (2) Conversely, suppose the geometric instance admits a perfect non‑crossing extension. The design of the clause gadget forces each clause to have exactly one active port; otherwise a crossing would be unavoidable. Hence the pattern of active ports defines a truth assignment that satisfies the 1‑in‑3‑SAT formula.

Thus the decision problem is NP‑hard, and since a candidate extension can be verified in polynomial time, the problem lies in NP, establishing NP‑completeness.

The authors also show that the same reduction works for bichromatic (red‑blue) matchings, where edges may only connect points of different colors. By coloring the points of each gadget appropriately, the reduction remains valid, proving NP‑completeness for the bichromatic version as well.

In the discussion, the paper contrasts this hardness result with the existence theorem for perfect non‑crossing matchings, highlighting that the presence of a partial matching dramatically changes the computational landscape. The authors suggest several avenues for future work: identifying special point configurations (e.g., points on a convex hull, grid points) where the augmentation problem might become tractable, designing approximation algorithms or fixed‑parameter algorithms, and exploring related augmentation problems such as augmenting non‑crossing spanning trees or planar graphs.

Overall, the paper provides a clean and elegant geometric reduction that settles the complexity of augmenting partial straight‑line matchings, extending the frontier of computational geometry by linking classic planar matching existence results with modern complexity theory.