Non-crossing Connectors in the Plane

We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R_1,…,R_n in the plane and finite point sets P_i subset of R_i for i=1,…,n, are there non-crossing connectors y_i for (R_i,P_i), i.e., arc-connected sets y_i with P_i subset of y_i subset of R_i for every i=1,…,n, such that y_i and y_j are disjoint for all i different from j? We prove that non-crossing connectors do always exist if the regions form a collection of pseudo-disks, i.e., the boundaries of every pair of regions intersect at most twice. We provide a simple polynomial-time algorithm if the regions are axis-aligned rectangles. Finally we prove that the general problem is NP-complete, even if the regions are convex, the boundaries of every pair of regions intersect at most four times and P_i consists of only two points on the boundary of R_i for i=1,…,n.

💡 Research Summary

The paper introduces the “non‑crossing connectors” problem, a geometric decision problem that asks whether, given n simply‑connected planar regions R₁,…,Rₙ and finite point sets Pᵢ⊆Rᵢ for each i, one can select for every i a connected set yᵢ satisfying Pᵢ⊆yᵢ⊆Rᵢ such that all yᵢ are pairwise disjoint. This formulation simultaneously generalizes planar embedding of multiple paths and the classic problem of connecting point pairs without crossings, and it has natural applications in network layout, VLSI routing, and geographic information systems.

The authors first focus on a structural restriction: the regions form a collection of pseudo‑disks, meaning that the boundaries of any two regions intersect at most twice. Under this condition they prove an existence theorem: for any such family of regions and any choice of point sets, a family of non‑crossing connectors always exists. The proof proceeds by an inductive construction that processes the regions in an order respecting the planar intersection graph. At each step a new connector is drawn inside the “free” part of its region, which is guaranteed to be a topological disk after the removal of previously placed connectors. By carefully choosing the connector to stay as close as possible to the boundary of the region, the authors show that no crossing with earlier connectors can occur. The argument relies on planar graph properties and Euler’s formula to maintain the invariant that the uncovered portion of each region remains simply connected.

Next, the paper addresses algorithmic tractability for a more concrete class of regions: axis‑aligned rectangles. Because rectangles intersect only in a grid‑like fashion, their intersection graph is a comparability graph that can be topologically sorted. The authors present a sweep‑line algorithm that processes rectangles in increasing x‑order, maintaining an interval tree for the currently active rectangles. For each rectangle, the algorithm connects its uncovered points using a minimal “ladder” shape (a union of a vertical and a horizontal segment) that stays inside the rectangle and avoids previously drawn connectors. Intersection tests are performed in O(log n) time, yielding an overall O(n log n) running time. The algorithm is simple, deterministic, and directly implementable, making it attractive for practical layout problems where the regions are naturally rectangular.

The final contribution is a hardness result for the unrestricted problem. Even when each region is convex, the boundaries of any two regions intersect at most four times, and each point set consists of exactly two points placed on the boundary of its region, the problem remains computationally intractable. The authors construct a polynomial‑time reduction from 3‑SAT. They design two types of gadgets: variable gadgets and clause gadgets, each realized as a convex polygon with at most four boundary intersections with other gadgets. Inside each gadget two points must be connected by a connector that encodes a truth assignment (e.g., a “true” connector runs on one side of the gadget, a “false” connector on the other). The geometry of the gadgets forces connectors from different gadgets to intersect unless the underlying truth assignment satisfies the corresponding clause. Consequently, a set of non‑crossing connectors exists if and only if the original Boolean formula is satisfiable. Since the problem is clearly in NP (a family of connectors can be verified in polynomial time), this establishes NP‑completeness.

In summary, the paper delineates a clear boundary between tractable and intractable instances of the non‑crossing connectors problem. For pseudo‑disk families, existence is guaranteed; for axis‑aligned rectangles, a constructive O(n log n) algorithm is provided; yet even modest relaxations—convex regions with limited boundary intersections—lead to NP‑completeness. These results not only advance theoretical understanding of planar connectivity under geometric constraints but also suggest practical algorithms for special cases and motivate further research on approximation, parameterized complexity, higher‑dimensional extensions, and dynamic settings.