Cyclic Pair Merging: Theory & Applications

• Cyclic pair merging is a combinatorial operation that iteratively merges paired elements within cyclic symmetric structures, facilitating classification and reduction in diverse systems.
• In neural network pruning, cyclic pair merging consolidates less significant weights into key parameters using curvature analysis, achieving substantial sparsification with minimal accuracy loss.
• It also enables explicit construction of pattern-avoiding permutations and streamlines discrete Morse function reductions, unifying combinatorial, algebraic, and computational approaches.

Cyclic pair merging is a combinatorial operation arising in several mathematical and algorithmic contexts: partition categories indexed by cyclic groups, efficient neural network pruning, pattern-avoiding permutations, and discrete Morse theory via generalized merge trees. The core principle is the iterative consolidation or "merging" of pairs—often arranged in a cyclic structure—subject to constraints determined by combinatorial, algebraic, or computational rules. This operation is employed both as a structural tool for classification and as an efficient mechanism for reduction in high-dimensional systems.

1. Cyclic Pair Merging in Two-Colored Pair Partition Categories

Cyclic pair merging was introduced in the classification of categories of two-colored pair partitions, notably in Mang–Weber's paper of partition categories indexed by cyclic groups (Mang et al., 2018). Consider a finite set split into upper and lower rows, with each point colored black or white and partitioned into disjoint pairs (blocks). A cyclic rotation operation removes the leftmost upper-row point, reattaches it (with swapped color) to the left of the lower row, then cyclically shifts all points.

The key objects are neutral pair partitions, where each block contains one white and one black point and has net color sum zero (the normalized coloring assigns $+1$ or $-1$ to each point depending on its position and color). Merging of blocks in these categories is realized by partition composition: stacking partitions and erasing the middle row causes circuits to merge, subject to preservation of neutrality.

Cyclic group actions (via rotation) index the categories $S_w$ through invariants: a pair partition $p$ lies in $S_w$ iff, for every block $B$, the color-imbalance between its legs (measured on a single line after cyclic rotation) is divisible by $w$; i.e., the sector color-sum $\sigma_p(S)$ for each block satisfies $\sigma_p(S) \in w\mathbb{Z}$. This cyclic symmetry strictly governs merging: only blocks compatible with modular divisibility merge, and thus categories $S_w$ are precisely classified by divisibility subgroups $w\mathbb{Z} \subset \mathbb{Z}$.

Examples are explicit: for $w=2$ (category $S_2$), only even color-gaps are permitted, while for $w=3$ ($S_3$), gaps must be divisible by three. The merging operation respects these modular constraints, forming the basis for a complete classification theorem: the only subcategories strictly between the trivial and maximal categories are indexed by these cyclic modular conditions.

2. Efficient Neural Network Pruning via Cyclic Pair Merging

The cyclic pair merging paradigm has been adapted in deep learning for efficient DNN pruning, notably as the foundation of the CAMP-HiVe algorithm (Uddin et al., 9 Nov 2025). Here, the operation consolidates weight parameters of a network model by cyclically merging "less significant" weights into "significant" weights, determined by curvature analysis.

Significance is computed via the dominant eigenvector of the layer-wise Hessian (approximated by power iteration and Hessian-vector products, avoiding full Hessian computation). Weights are partitioned into "significant" ($S$) and "less significant" ($L$) sets using percentile thresholding; each weight in $L$ is cyclically mapped to a weight in $S$ (indexing by modulo $|S|$), its value added, and then zeroed. This procedure minimizes the second-order increase in loss ($\Delta\mathcal L$ via Taylor expansion) and yields aggressive sparsification with minimal accuracy loss.

The algorithm pseudocode is structured as follows:

for layer in network: compute Hessian-vector dominant eigenvector v by power iteration significance = abs(v) threshold = percentile(significance) S, L = partition weights by threshold for i in L: j = (i % |S|) + 1 W_S[j] += W_L[i] # merge W_L[i] = 0 # prune

Dynamic curvature adaptation is integral: significance scores are recomputed after each pruning/fine-tune cycle to reflect changing model priorities. Empirical benchmarks demonstrate substantial FLOPs and parameter reduction, with accuracy maintained or improved across datasets and architectures. This cyclic merging with Hessian-based significance outperforms classical magnitude-, filter-, and traditional second-order pruning methods, yielding improved efficiency on resource-constrained systems.

3. Cyclic Pair Merging in Pattern-Avoiding Permutations

Archer–Graves–Laudone introduced cyclic pair merging to construct large pattern-avoiding cycles from smaller ones (Archer et al., 7 May 2025). Consider $C_n(\sigma)$: the set of $n$-cycles in $\mathcal{S}_n$ avoiding a fixed pattern $\sigma \in S_3$. A partial groupoid structure is defined on the disjoint union $C(\sigma)$, with binary operations $\otimes_\sigma$ (merge) defined by explicit formulas parameterized by pattern-specific relabeling rules.

For example, for $\sigma = 312$, the merge operation $\alpha * \beta$ concatenates two cycles, shifts entries of $\beta$ to accommodate the size of $\alpha$, and appends special elements to preserve cyclicity, ensuring the resulting cycle remains $\sigma$-avoiding.

Pseudocode for the $\sigma=312$ merge:

Input: α=a₁…a_{m−2}21, β=b₁…bₙ Output: γ∈C_{m+n−2}(312) γ[1…m−2] ← (a₁,…,a_{m−2}) for i=1 to n: if bᵢ=2 then γ[m−2 + i] ← 2 else γ[m−2 + i] ← bᵢ + (m−2)

Recursive lower bound theorems for $c_n(\sigma)$ (the quantity $|C_n(\sigma)|$) are obtained by partitioning into disjoint merge images indexed by finite prefix sets $A_k$. Explicit computer enumeration yields rigorous growth rates (e.g., $c_n(312) > 3.0438^n$, $c_n(321) > 3.1788^n$). The double-augmentation argument proves $c_n(\sigma) \ge 2 c_{n-1}(\sigma)$ for all $\sigma \in S_3 \setminus \{123\}$.

4. Cyclic Pair Merging in Generalized Merge Trees and Morse Theory

Brüggemann–Scoville extend classical merge trees to "generalized merge trees" (GMT), which track not only component merges but also births of cycles as unary nodes (Brüggemann et al., 2023). The cyclic pair merging operation in this context involves canceling a cycle node together with its paired critical event, preserving the GMT axioms, and simplifying the Morse function on the underlying graph.

Construction proceeds by traversing the critical edges of a finite 1-dimensional CW complex (graph), alternately attaching unary (cycle) and binary (merge) nodes labeled in Morse order. The cyclic pair merging operation contracts a singleton branch in the tree, updating labels and children, and corresponds precisely to discrete Morse function cancellation of a 1-cell with a 0-cell.

The framework establishes a bijection between component–merge equivalence classes of discrete Morse functions and isomorphism classes of GMTs. The cyclic pair merging move provides both an algorithmic tool and a combinatorial criterion for function equivalence and optimal reduction. The inverse problem—characterizing all Morse functions from GMT data—is solved by enforcing chirality, labeling, and minimum conditions on the tree structure.

5. Theoretical and Practical Significance

Cyclic pair merging provides a unifying combinatorial formalism across several domains:

• In partition categories, it offers exhaustive classification via modular constraints indexed by cyclic group actions.
• For DNN pruning, it enables curvature-aware, high-efficiency compression strategies with strong empirical and computational guarantees.
• In permutation pattern avoidance, it facilitates explicit recursive construction and growth-rate analysis of cycle-avoiding sets.
• In Morse theory on graphs, it yields structural equivalence, effective cancellation mechanics, and full solution to the inverse reconstruction problem.

A plausible implication is that cyclic pair merging, as an algebraic and algorithmic primitive, can be generalized or hybridized into frameworks for other domains where structure-preserving reduction or merge operations are essential. The role of cyclic symmetries, modular indexings, and local-to-global compatibility is central to all its instantiations.

6. Connections, Controversies, and Extensions

No significant controversies are presently associated with cyclic pair merging, though some open directions remain:

• In two-colored pair partitions, the second family of partition categories not indexed by cyclic groups awaits full classification (Mang et al., 2018).
• In neural pruning, the precise interaction of cyclic mergers with more complex invariance or structured sparsity constraints offers further avenues for exploration.
• Generalized merge trees may be further enriched to support multidimensional Morse theory or adapted to persistent homology in TDA, extending the theoretical reach of cyclic pair merging mechanics.

The synthesis of modular group actions, combinatorial block merging, importance ranking, and label-preserving reduction constitutes the backbone of cyclic pair merging, positioning it as a fundamental operation in modern combinatorial, algebraic, and computational theory.