Nonconvex Polygonal Partitions

• Nonconvex polygonal partitions are decompositions of polygonal domains into non-overlapping pieces, accommodating concavities that convex partitions cannot handle.
• Recent algorithmic advances leverage dynamic programming and Steiner points to efficiently partition complex geometries despite high computational complexity.
• These partitions are fundamental in mesh generation, motion planning, and numerical simulations, enabling robust and efficient solutions in practical applications.

A nonconvex polygonal partition refers to a decomposition of a polygonal domain—potentially possessing arbitrary concavities—into a finite collection of (not necessarily convex) polygonal pieces that jointly cover the original domain and have pairwise non-overlapping interiors. Such partitions are foundational across computational geometry, mesh generation, computational vision, finite element analysis, and motion planning, especially when convexity is unattainable or undesirable due to geometric or application-driven constraints.

1. Fundamental Definitions and Partition Classes

Let $P \subset \mathbb{R}^2$ denote a simple (or multiply-connected) polygonal domain. A partition $\mathcal{Q} = \{ Q_1, ..., Q_m \}$ is a set of connected polygonal subsets such that:

• $\bigcup_{i=1}^m Q_i = P$
• $$\mathrm{int}(Q_i) \cap \mathrm{int}(Q_j) = \emptyset$$ for all $i \neq j$

Partitions are typically required to be weakly-simple, meaning pieces may overlap on measure-zero sets but are otherwise “separable” by negligible perturbations (Abrahamsen et al., 2022). Partition pieces may freely introduce new vertices (Steiner points) or be restricted to only the original vertices of $P$.

Within this general modeling, several crucial specializations exist:

• Star-shaped partitions: Each piece $Q$ is star-shaped, i.e., $\exists p \in Q$ such that $[p, q] \subset Q$ for all $q \in Q$ (Abrahamsen et al., 2023, Obermeyer et al., 2010).
• Convex partitions: Each $Q$ is convex. Convex partitions have dominated much classical literature but are more restrictive than star-shaped or general nonconvex polygonal decompositions.
• Axis-aligned rectangular partitions: For rectilinear $P$, one may require each $Q$ to be a rectangle; this introduces additional geometric and combinatorial structure (Kim et al., 2021).
• Partitions with bounded size: Pieces are constrained by diameter, perimeter, or containment within specified shapes (square, disk, etc.) (Abrahamsen et al., 2022).
• Partitions by function (e.g., minimum-ink, maximum-thickness, min-area cover): Optimization is with respect to a global functional over pieces, subject to various constraints.

The core challenge arises from nonconvexity, as the geometry precludes global monotonicity or directional decomposition, necessitating more intricate combinatorial and geometric methods.

2. Algorithmic Methods for Nonconvex Polygonal Partitioning

2.1. Minimum Star-shaped Partition (General Simple Polygon)

A longstanding open problem—minimum star-shaped partition of a simple polygon—was recently resolved with a polynomial-time algorithm. Let $P$ be an $n$-vertex simple polygon. The goal is to compute a partition of $P$ into the smallest number of star-shaped pieces. The approach (Abrahamsen et al., 2023) uses:

• Tripods: Combinatorial structures linking three star centers and their “support” (concave vertices), yielding $O(n^6)$ candidate star center locations.
• Candidate Steiner points: Generated via lexicographically maximal area consideration; every corner of a piece either lies on a star center-to-vertex sight line or at their intersection.
• Separator-based dynamic programming: The space of possible separators—short $(B_1\!-\!A_1\!-\!B_2)$ and long $(B_1\!-\!A_1\!-\!Z\!-\!A_2\!-\!B_2)$—is polynomially bounded after candidate enumeration, and a dynamic program merges subproblems according to six “transition” types. The overall complexity is $O(n^{105})$.

Steiner points are essential: solutions without them may require $\Theta(n)$ pieces even if two pieces suffice given interior star centers.

2.2. Partitions by Bounded Size (General Polygon)

A two-phase approximation framework allows partitioning a simple polygon $P$ into $O(1)$-approximate minimal number of pieces, each meeting one of six geometric constraints: containment in an axis-aligned or arbitrarily rotated unit square, unit disk, straight-line or geodesic diameter at most 1, or perimeter at most 1 (Abrahamsen et al., 2022). The algorithmic structure is:

• Boundary phase: Recursively covers $\partial P$ with as few “boundary pieces” as possible, each compliant with the constraint, yielding at most $2\psi^*-1$ boundary pieces (where $\psi^*$ is the optimal number of pieces).
• Interior phase: Decomposes the remaining interior using a grid (with squares, disks, etc.), or by further splitting into “subfields” in geodesic-diameter and perimeter-constrained variants.
• Approximation ratio: The algorithms guarantee solution sizes at most constants (13–3728, depending on constraint) times the optimum.

Allowing Steiner points is vital both for existence of a solution and for tight approximation guarantees.

2.3. Rectangular Partitions of Rectilinear Polygons

For rectilinear polygons, minimum-ink and maximum-thickness partitions can be realized via a dynamic programming paradigm exploiting the canonical grid structure (Kim et al., 2021):

• Minimum ink partition: Given $n$-vertex $P$, one achieves $O(n^3)$ total time and $O(n^2)$ space using recursive decomposition by uni-rectangle cuts and careful prefix-sum structures.
• Maximum thickness partition: Identifies the rectangular partition with the largest minimum side among pieces; implemented via dynamic programming with segment-tree-based structures for side length queries, yielding $O(n^3 \log^2 n)$ time.
• With holes: Hardness arises; the thick-partition decision problem (given threshold $\delta$ and maximum number of pieces $k$) is NP-complete when holes are present.

3. Visibility-based and Distributed Nonconvex Partitions

In visibility coverage and robotics, nonconvex polygonal partitions naturally arise by decomposing the environment into cells each seen from a single vantage—typically a star-convex cell (Obermeyer et al., 2010):

• Incremental visibility partitioning: Builds a tree of star-convex cells using local visibility graphs. At each step, unresolved “gap edges” spur new cells associated with new vantage points.
• Distributed deployment: The partition algorithm requires only line-of-sight sensing and peer-to-peer agent communication, making the scheme fully distributed and robust to unknown environments.
• Theoretical guarantees: For an environment with $n$ vertices and $h$ holes, at most $n+2h-2$ cells suffice, with the union covering the environment and the vantage points forming a connected visibility graph.

This decentralized approach fundamentally differs from triangulation or classical convex decompositions, as it builds the partition incrementally and locally.

4. Applications in Mesh Generation and Numerical Methods

Nonconvex polygonal partitions are indispensable in advanced mesh generation, especially for finite element and finite volume methods in computational PDEs. Of particular recent interest is the use of nonconvex partitions (often with arbitrary or adaptive polygonal elements) in weak Galerkin finite element methods (Wang et al., 21 Dec 2025):

• Shape-regular nonconvex polygonal partition: Requires each polygon be star-shaped with respect to a ball proportional to its diameter and have edge lengths bounded below, but permits arbitrary concavities (excluding pathological needles or slivers).
• Local polynomial approximation: On each nonconvex polygon, polynomial function spaces and edge traces are defined, with degrees tailored depending on convexity/nonconvexity.
• Discrete weak gradient: Crucially, a stable discrete gradient operator is defined on possibly highly concave polygons using integration-by-parts relationships and appropriate function spaces.
• System properties and error estimates: The resulting global linear system is symmetric and positive definite; optimal $H^1$ error convergence of $O(h^k)$ is attained even for complex nonconvex meshes.

This demonstrates that nonconvex polygonal partitions are not merely an artifact of geometric decomposition but can serve as robust foundations for highly accurate and stable simulation algorithms.

5. Optimization, Hardness, and Covering Results

Further refinement of nonconvex polygonal partitions considers global covering and optimization functionals, generalizing classical art gallery and guarding problems:

• Minimum-area convex covering: Given a family of disjoint input polygons, the process of convexification—repeatedly merging overlapping convex hulls—yields a unique optimal covering by disjoint convex polygons, with minimum summed area. These can be computed in near-linear time with advanced dynamic data structures (Dahlhaus et al., 2019).
• NP-hardness and open problems: Many nonconvex partition problems become hard with topological complications (holes) or added constraints on piece structure. For example, axis-aligned square partitioning is NP-hard for polygons with holes (Abrahamsen et al., 2022); thick rectangular partitioning is likewise NP-complete (Kim et al., 2021).
• Comparison with no-Steiner partitions: In high-concavity scenarios, partitions restricted to original $P$ vertices may require $\Omega(n)$ pieces, while Steiner-point-enabled partitions reduce to $O(1)$. Chord-only and no-Steiner restrictions often preclude existence or efficient approximation (Abrahamsen et al., 2022).

6. Practical and Theoretical Implications

Nonconvex polygonal partitions have diverse and significant consequences:

• CNC machining: Minimum star or convex partitions correspond to regions that can be milled without retracting the tool, directly minimizing process time (Abrahamsen et al., 2023).
• Motion planning: Star-shaped or visibility partitions align with roadmap structures, simplifying cell-by-cell traversal or localization (Obermeyer et al., 2010).
• Mesh adaptation in simulation: The ability to use general nonconvex elements enhances flexibility in capturing geometric features, reducing element count, or aligning with solution structure (Wang et al., 21 Dec 2025).
• Manufacturing and biomedicine: Small-size-bounded partitions are integral to planning, collision detection, and sample extraction (Abrahamsen et al., 2022).
• Visualization and shape parameterization: Nonconvex partitions, especially star-shaped or bounded-diameter, facilitate morphing and mesh parameterization (Abrahamsen et al., 2023).

Despite their geometric complexity, modern theory and algorithmics permit both rigorous approximation guarantees and efficient, robust computation for a wide range of fundamental and applied problems involving nonconvex polygonal partitions.