Optimal Polyhedral Region Merging

• The paper presents rigorous frameworks that combine dynamic programming, polyhedral approximations, and statistical tests to achieve global optimality.
• It explains various algorithmic paradigms such as DRM, OA, and symmetry reduction that merge adjacent regions while controlling approximation errors.
• Applications in image segmentation, neural network compression, and mixed-integer programming demonstrate scalable solutions with strong theoretical guarantees.

Optimal polyhedral region merging is a mathematical and algorithmic concept underpinning a broad class of problems in computational geometry, combinatorial optimization, image processing, mixed-integer convex programming, network information theory, statistical estimation, and structure-preserving neural network compression. The term "optimal" indicates a global criterion—minimizing cost, maximizing fidelity, or achieving minimal description—subject to polyhedral region constraints, with "merging" referring to the union or aggregation of adjacent polyhedral domains while enforcing specified properties or bounds. This process is essential in applications requiring efficient partitioning, compact region description, computational tractability, and rigorous control over approximation error.

1. Mathematical Foundations and Definitions

At its core, polyhedral region merging involves operations on sets represented as intersections of finitely many halfspaces or their combinatorial assemblages. A polyhedron $\mathcal{P}$ in $\mathbb{R}^n$ is characterized as: $\mathcal{P} = \{ x \in \mathbb{R}^n : A x \le b \}$ for $A \in \mathbb{R}^{m\times n}$, $b \in \mathbb{R}^m$. Merging adjacent regions $\{\mathcal{P}_i\}_{i=1}^k$ seeks to produce a union $\mathcal{P}_{\text{merged}}$ optimally, often under an objective functional $F(\{\mathcal{P}_i\})$ (e.g., minimization of a cost or maximization of mutual consistency).

Polyhedral region merging is optimal if—subject to application-specific predicates, constraints, or error bounds—the final merged region(s) cannot be further refined (no mergeable regions left) nor trivially split (no over-merged regions), as formalized in image segmentation (Peng et al., 2010), KAN compression (Zhang, 5 Oct 2025), and d.c. optimization (Dahl et al., 2019).

2. Algorithmic Paradigms for Region Merging

A variety of algorithmic paradigms have been developed for optimal merging:

• Dynamic Region Merging (DRM): In image segmentation, DRM operates on an initial over-segmentation (e.g., superpixels from watershed), iteratively merging the most similar (least dissimilar) neighboring regions subject to a statistical predicate combining minimum edge weight and SPRT-based consistency (Peng et al., 2010). The process is formalized as a dynamic programming minimization over transition costs:

$$F = \sum_i F_i, \qquad \text{with } F_i \text{ based on minimal edge weights}$$

Merging terminates when the predicate is unsatisfied for all adjacent region pairs, preserving both under- and over-merging global invariants.

• PolyKAN DP Compression: In KAN neural networks, each input dimension is partitioned by spline knots into axis-aligned polyhedral regions (Zhang, 5 Oct 2025). Optimal compression involves merging contiguous spline regions if their union admits a joint polynomial approximation within an $\epsilon$ bound. The DP recurrence per spline is:

$$dp[i] = \min_{0 \leq j < i} \{ dp[j] + 1 \;\; | \;\; \text{CheckMergability}(T, j, i, \delta) \}$$

with global guarantees on compressed model fidelity.

• Outer Approximation (OA): In mixed-integer convex optimization, polyhedral outer approximations are iteratively refined by tangent (linearized) cuts generated at solutions of convex subproblems, leading to global solution discovery (Lubin et al., 2016). Extended formulations in higher-dimensional space drastically reduce iteration counts, and DCP modeling supports automation.
• Symmetry-Exploiting Polyhedral Projection: For network information theory, region merging is formulated as polyhedral projection—eliminating redundant dimensions and merging symmetric faces for a minimal description (Apte et al., 2016). Algorithms such as symCHM utilize group-theoretic symmetry reduction to minimize computational burden.
• Concave Minimization for d.c. Optimization: Polyhedral difference-of-convex (d.c.) problems can be reformulated as concave minimization over polyhedral domains, solving for optimal merging under a global criterion (Dahl et al., 2019). Existence and solution are certified via domain and recession cone conditions.

3. Statistical and Consistency Criteria

Optimal region merging often depends on statistical consistency predicates rigorously quantifying the "mergeworthiness" of regions:

• Sequential Probability Ratio Test (SPRT): In DRM, SPRT is used to accumulate evidence (log-likelihood ratios) supporting or refuting consistency between neighboring regions:

$\delta = \sum_i \log \frac{P(x_i|H_1)}{P(x_i|H_0)}$

where $H_0$ is the hypothesis of inconsistency and $H_1$ is consistency; merging proceeds if $\delta$ exceeds predetermined thresholds $A$ or $B$ (Peng et al., 2010).

• Error Control in PolyKAN: KAN compression predicates on the existence of a joint polynomial approximator for merged regions $R_i\cup R_j$ such that

$$\max_{x \in R_i \cup R_j} |p_{\text{orig}}(x) - p_{\text{merged}}(x)| \le \epsilon$$

ensuring compressed model outputs remain within desired fidelity bounds (Zhang, 5 Oct 2025).

• Risk Bounding in Polyhedral Estimates: In estimation problems, risk is bounded by decomposing contributions from ellitopic and polyhedral components of the constraint set, exploiting LMIs and semidefinite relaxation, with tight bounds formulated for aggregate merged contrasts (Juditsky et al., 2022).

4. Structural and Computational Properties

Polyhedral region merging algorithms emphasize:

• Graph-based Acceleration: Utilizing Nearest Neighbor Graphs (NNG), DRM identifies mutually nearest cycles for candidate region merges, achieving significant speedup over exhaustive adjacency graph scans (Peng et al., 2010).
• Extended Formulations and Conic Representations: Strengthening polyhedral approximations in a lifted space via auxiliary variables (epigraphs) and conic duality delivers robust, general models with far fewer constraints (Lubin et al., 2016).
• Symmetry Inference: Group-theoretic symmetry reduction, as in symCHM, automatically merges redundant polyhedral faces, vertices, and constraints, resulting in a minimal (optimal) outer region representation for rate regions (Apte et al., 2016).
• Sequential Optimization: Polyhedral restrictions in nonconvex optimization (e.g., AC OPF feasibility) permit guaranteed feasible solutions via affine constraints, with iterative refinement yielding successively tighter inner bounds (Christianen et al., 2023).

5. Applications and Examples

The concept underlies several fields:

• Image Segmentation: DRM yields high-quality segmentations—object regions with well-preserved boundaries—demonstrated on natural image datasets with competitive F-measure scores (up to 0.66) versus human annotation (0.79) (Peng et al., 2010).
• Neural Network Compression: PolyKAN provides the first provable minimal compression of KANs under error bounds, crucial for resource-constrained deployment and interpretability (Zhang, 5 Oct 2025).
• Mixed-Integer Convex Programming: OA algorithms with extended polyhedral approximations enable solving many previously intractable benchmarks (reducing iteration count from 2,685 to 994 across MINLPLIB2) (Lubin et al., 2016).
• Network Coding: Polyhedral projections yield optimally merged characterizations of rate regions, crucial for converse proofs and symmetric problem instances (Apte et al., 2016).
• Statistical Estimation: Polyhedral estimates with optimized contrast aggregation reduce estimation risk in linear inverse problems, especially when the signal set combines ellitopic and polytope geometries (Juditsky et al., 2022).
• Power Systems: Polyhedral restrictions provide feasible inner approximations in OPF, computationally efficient and theoretically guaranteed for realistic grid operation (Christianen et al., 2023).

6. Theoretical Guarantees and Limitations

Optimal polyhedral region merging frameworks rigorously establish:

• Global Optimality and Stopping Criteria: Merging terminates at a globally optimal configuration; theoretical proofs confirm absence of both under- and over-merging (Peng et al., 2010, Zhang, 5 Oct 2025).
• Polynomial-time Complexity: Dynamic programming approaches in PolyKAN ensure tractable (polynomial) time solutions across network width, depth, and spline knots (Zhang, 5 Oct 2025).
• Error Propagation Bounds: Error control analysis ensures merged models (image segmentation, neural networks, estimation routines) do not exceed prescribed approximation or risk levels (Zhang, 5 Oct 2025, Juditsky et al., 2022).
• Scalability: Extended formulations and symmetry-based reduction techniques maintain scalability for high-dimensional, complex polyhedral instances (Lubin et al., 2016, Apte et al., 2016).

Limitations arise from the conservative nature of polyhedral restrictions in feasibility approximations (e.g., OPF) and performance dependencies on chosen predicates/parameters (e.g., consistency thresholds or error allocation schemes).

7. Broader Impacts and Future Directions

Optimal polyhedral region merging advances the development of mathematically grounded, efficient algorithms in computational geometry, signal processing, convex and combinatorial optimization, network coding, and model compression. The formalization of merging as a DP or OA problem, incorporation of statistical/structural predicates, and exploitation of symmetry lay the foundation for rigorous, scalable, and interpretable solutions in domains where polyhedral structures are intrinsic.

Active directions include deeper theoretical characterization of error allocation, tighter integration of structural properties (e.g., topology preservation in image vectorization (He et al., 24 Sep 2024)), extension to semidefinite and non-polyhedral convex sets, unified frameworks for hybrid geometric models, and adaptive merging strategies in data-driven settings.

In sum, optimal polyhedral region merging constitutes a foundational methodology for achieving compact, efficient, and provably correct aggregation of polyhedral domains across sciences and engineering.