[Paper Review] The Partition Spanning Forest Problem

Philipp Kindermann, Boris Klemz+3 more|arXiv (Cornell University)|Sep 7, 2018

Computational Geometry and Mesh Generation5 references4 citations

TL;DR

This paper investigates the computational complexity of the Partition Spanning Forest Problem, where the goal is to draw a crossing-free spanning forest such that each tree contains exactly one color class of points. It proves the problem is NP-complete when each color class has up to five points, but solvable in O(n²) time when each class has at most three points; it further shows NP-completeness for the linear forest variant even with four points per class.

ABSTRACT

Given a set of colored points in the plane, we ask if there exists a crossing-free straight-line drawing of a spanning forest, such that every tree in the forest contains exactly the points of one color class. We show that the problem is NP-complete, even if every color class contains at most five points, but it is solvable in $O(n^2)$ time when each color class contains at most three points. If we require that the spanning forest is a linear forest, then the problem becomes NP-complete even if every color class contains at most four points.

Motivation & Objective

• To determine the computational complexity of the Partition Spanning Forest Problem under bounded color class sizes.
• To investigate whether the problem remains tractable when each color class contains at most three or four points.
• To analyze the complexity of the variant where the spanning forest must be a linear forest (i.e., each component is a path).
• To establish tight complexity bounds by proving NP-completeness for specific parameterized cases.
• To provide a polynomial-time algorithm for the case of at most three points per color class using 2-SAT formulation.

Proposed method

• Formulate the problem as a 2-SAT instance when each color class has at most three points, encoding connectivity and non-crossing constraints.
• Use geometric constraints and convex hull intersections to enforce variable truth assignments in variable and wire gadgets.
• Construct specialized gadgets—variable, wire, splitting, inverter, and OR-gadgets—for a reduction from Planar 3-SAT.
• Design clause and OR-gadgets to simulate logical satisfaction in the geometric drawing context.
• Prove correctness of the reduction by showing that a valid geometric drawing exists if and only if the underlying Planar 3-SAT formula is satisfiable.
• Use planar embedding and geometric realizability to ensure that all constraints (non-crossing, connectivity) are preserved in the construction.

Experimental results

Research questions

• RQ1 Is the Partition Spanning Forest Problem solvable in polynomial time when each color class contains at most three points?
• RQ2 What is the complexity of the problem when each color class contains up to four or five points?
• RQ3 Is the problem NP-complete for the linear forest variant, even when each color class has at most four points?
• RQ4 Can the problem be reduced to Planar 3-SAT via geometric gadgetry while preserving non-crossing and connectivity constraints?
• RQ5 What is the complexity status of the problem when each color class has exactly four points?

Key findings

• The Partition Spanning Forest Problem is solvable in O(n²) time when each color class contains at most three points, via a 2-SAT formulation.
• The problem is NP-complete when each color class contains at most five points, even with general point position.
• The linear forest variant (where each tree is a path) is NP-complete even if each color class contains at most four points.
• The complexity remains open for the non-linear case when each color class contains exactly four points.
• A polynomial-time reduction from Planar 3-SAT to the geometric problem establishes NP-completeness using custom geometric gadgets.
• The construction ensures that a valid crossing-free drawing exists if and only if the original Planar 3-SAT formula is satisfiable.