Polygon Matching: Methods & Applications

• Polygon Matching Strategy is a formal approach that employs invariant techniques, combinatorial algorithms, and dynamic programming to align and compare polygonal structures.
• The method integrates matrix-product enumeration and affine invariants to achieve efficient, noise-resilient matching and robust shape analysis across various applications.
• It is applied in fields such as computer vision, combinatorics, and blockchain analytics, offering theoretical guarantees and practical solutions for complex matching problems.

A polygon matching strategy is a formal or algorithmic approach for determining correspondences, similarities, bijections, or optimal alignments between polygons or between polygons and other mathematical objects. Such strategies are central to fields spanning graph theory, computational geometry, combinatorics, shape analysis, computer vision, mesh processing, and blockchain analytics. Key approaches incorporate combinatorial algorithms, invariant-based techniques, optimization methods, and enumeration principles, each suited to specific theoretical or application settings.

1. Foundational Concepts and Problem Settings

Polygon matching problems are diverse, ranging from geometric shape correspondence and graph embedding to combinatorial object bijections and the enumeration of matchings in polygonal structures. Central objects of paper include:

• Polygons as curves or point sets (for non-rigid matching and computer vision)
• Polygonal graphs: planar graphs constructed from cycles (rings or chains)
• Polygon partitions: decompositions via non-crossing diagonals
• Polygon mesh boundaries: sequences of features (valence, curvature) for mesh selection
• Polygonal encodings in combinatorics: as compositions or Dyck paths
• Transaction flows between blockchains with “polygon” as chain name

These formulations induce a wide variety of matching paradigms, from geometric alignment under transformation groups to discrete combinatorial bijections and trace formulas governed by structural constraints.

2. Enumeration and Algebraic Approaches: Maximal Matchings in Polygon Rings

A prominent strategy in mathematical chemistry and combinatorics is the enumeration of maximal matchings in polygon rings, as detailed for hexagonal (and arbitrary) polygon rings (Li et al., 10 Jun 2025). In this setting, a maximal matching is an edge subset that cannot be expanded further.

The algorithm employs matrix-product and trace formulas:

$$\Psi(Z) = \mathrm{tr}\left( \prod_{i=1}^n T_{(s_i,k_i)} \right)$$

where each $T_{(s_i,k_i)}$ is a $9 \times 9$ transition matrix determined algorithmically for the local ring structure. The procedure decomposes the structure into primitive faces, computes their state transition matrices by enumerating all legal local matchings, and propagates boundary conditions globally via matrix multiplication. This dynamic programming via matrices leverages locality to achieve efficient and extensible counting, circumventing explicit enumeration for large structures.

This approach generalizes to arbitrary polygonal rings using the “trace of matrix product” paradigm, with explicit handling for open chains by vector contraction at boundary faces. The method is efficient, theoretically sound, and broadly applicable to the enumeration of chemical graph invariants in benzenoid hydrocarbons and related classes.

3. Invariant-Based and Affine/Similarity Matching

For geometric matching under high degrees of freedom (e.g., similarity or affine transformations), the strategy described in (Chávez et al., 2013) utilizes complex-analytic invariants:

$$\varphi_{n,j}(z_1,\ldots,z_n) = \frac{\sum_{k=1}^n \lambda_n^{jk} z_k}{\sum_{k=1}^n \lambda_n^{-jk} z_k}$$

where $\lambda_n = e^{2\pi i/n}$, yielding invariance under similarity transformations. Hashing powers $\varphi_{n,j}^n$ resolves ambiguities due to cyclical vertex orderings.

Efficient matching is achieved by precomputing these invariants for large collections and using multiple invariants for collision reduction, yielding query performance independent of collection size. For known affine transformations, affine-invariant pseudo-distances enable sublinear-time retrieval. The technique also extends to triangles under noise, with explicit error bounds in the invariant space.

These strategies are algorithmically robust, leveraging group actions and polynomial-time hashing to facilitate large-scale geometric indexing, and have theoretical guarantees under both clean and noisy conditions.

4. Structural and Dynamic Programming Methods for Non-Rigid and Part-Based Matching

Non-rigid polygonal matching—crucial for shape correspondence in computer vision, animation, and biomedical settings—demands flexible algorithms capable of handling: non-rigid deformations, partial occlusion, and clutter. The multi-part shape matching algorithm in (Litany et al., 2020) frames the search as a global optimization over possible segmentations and matchings, minimizing an energy function that includes partwise dissimilarity, structural (segmentation) penalties, and unmatched-part terms:

$$E(\mathcal{M}, S, T) = \sum_{(i,j) \in \mathcal{M}} D(S_i, T_j) + \lambda_1 C_{seg}(S,T) + \lambda_2 C_{unmatched}(S,T)$$

Optimization is performed via dynamic programming or branch-and-bound, with segmentation variables coupled directly into the matching process. Partial matches arise naturally due to the combinatorial matching of subset segments, and dense correspondences per matched part are resolved with tools like DTW or spline warping.

Empirical results show robustness under large, non-affine deformations, resilience to missing data, and clutter suppression, consistently outperforming rigid or single-part non-rigid baselines.

5. Polygon Matching in Combinatorics: Bijections and Enumerative Equivalences

Polygon matching strategies in enumerative combinatorics focus on bijections and cardinality-preserving transformations between polygon partitions, colored compositions, and Dyck paths (Gil et al., 7 Mar 2024). The key technique is the construction of part-preserving bijections that match building blocks—such as colored regions in a partition, primitive blocks in Dyck paths, or parts in a composition—by explicit marking and merging procedures.

For example:

• Decomposition of polygon partitions by non-crossing diagonals aligns each region with a part of the composition.
• Marking strategies equate cardinalities, e.g., splitting Dyck words at valleys to produce blocks matched to composition parts.
• Colorings propagate bijectively by assigning each size-$j$ building block one of $y_j$ possible colors.

These strategies yield explicit enumerative identities for the number of polygon partitions with coloring constraints and are algorithmically constructive, with enumeration formulas involving partial Bell polynomials and binomial coefficients. The approach provides a combinatorial toolkit for translating problems across representation domains.

6. Polygon Mesh and Shape Matching via Generalized Sequence Alignment

Mesh selection strategies for matching a given planar shape to a mesh catalogue are formulated as weighted Longest Common Subsequence (LCS) problems (Surynkova, 2016). Input boundaries are encoded as geometric feature sequences, and mesh boundary vertices carry valence and neighborhood information. Weighted lattices specify the matching utility for each feature pair. The optimal score is computed via a generalized LCS dynamic programming algorithm, permitting deletion in the user shape sequence and rotation in mesh sequences.

This methodology ensures polynomial-time optimality with respect to the specified objective and provides a basis for further parameterization and mesh adaptation. By abstracting mesh comparison to sequence alignment with geometric-aware weights, the approach enables robust and flexible shape-to-mesh associations.

7. Matching in Game and Dynamic Settings: NP-Hardness and Heuristic Strategies

Polygon matching arises in combinatorial games and dynamic contexts, as exemplified by the Dots & Polygons game (Buchin et al., 2020). Here, optimal matching—selecting maximal collections of vertex-disjoint cycles to maximize area—is equivalent to NP-hard cycle packing problems. For point sets in convex position, greedy heuristics (immediately claiming any available polygon/area, otherwise minimizing the area the opponent can later claim) yield provable guarantees: for small $n$, the last player can always win by forcing "easy endgames" where each move splits the area symmetrically. For larger or non-convex sets, the problem remains algorithmically intractable.

This domain-specific interpretation highlights that even playful or applied contexts inherit the algorithmic and computational hardness of their underlying polygon matching subproblems.

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Polygon matching strategies encompass a range of algorithmic and combinatorial techniques, from invariant-based indexing, matrix product enumeration, and global optimization to explicit combinatorial bijections and generalized sequence alignment. These methods are employed according to the constraints and goals of the application domain—be it robust shape correspondence under transformations, enumeration of combinatorial structures, efficient catalogue selection, or secure traceability in cross-chain blockchain analytics. Each paradigm leverages domain structure and mathematical invariants to achieve computational efficiency, robustness to noise or missing data, and, where possible, theoretical optimality.