Balanced k-Fold Assignment

• Balanced k-Fold Assignment is a technique that partitions datasets into k near-equal groups, preserving statistical properties and diversity across folds.
• The approach integrates combinatorial optimization, integer programming, and algorithms like the Hungarian method to achieve reproducible and scalable data splitting.
• Practical implementations include methods such as ABA for anticlustering and BIBD for peer review, supporting applications in cross-validation, clustering, and experimental design.

Balanced $k$-fold assignment concerns the partitioning of a dataset into $k$ disjoint, equal- or near-equal-sized groups (“folds”), under various constraints for reproducibility, statistical balance, pairwise coverage, or diversity. Central applications include model assessment via $k$-fold cross-validation, balanced clustering, stratified sampling, anticlustering, peer review assignment, and experimental design. The mathematical specification and algorithmic realization of balanced $k$-fold assignment integrate elements from combinatorial optimization, integer programming, linear assignment, and combinatorics.

1. Formal Problem Definitions

A balanced $k$-fold assignment involves allocating a set $X=\{X_1,\dots,X_N\}$ (typically in $\mathbb{R}^d$) into $k$ disjoint groups (“folds”) $F_1,\dots,F_k$ such that:

• Each fold $F_i$ receives either $n_i = \lfloor N/k\rfloor$ or $n_i = \lceil N/k\rceil$ points and $\sum_i n_i = N$.
• For stratified or multiclass settings, the marginal distribution of classes (or other properties) in each fold mirrors the global proportions as closely as possible, formalized by nonnegative integer matrices $F\in\mathbb{N}^{k\times C}$ with column and row sums matching per-class and per-fold totals.
• In clustering or anticlustering, further objectives are imposed: e.g., minimize mean square error (clustering), maximize within-fold diversity (anticlustering), maximize pairwise coverage (block designs).

Two central assignment formulations arise:

• Balanced $k$-means clustering: Assignments

$$\min_{Z,C}\; \frac{1}{n}\sum_{i=1}^n\sum_{j=1}^k z_{ij}\,\|X_i - C_j\|^2$$

subject to binary assignments $z_{ij}\in\{0,1\}$ and fixed cluster sizes $\sum_{i=1}^n z_{ij} = n_j$ (Malinen et al., 27 Jan 2025).

• Euclidean anticlustering (“max diversity”):

$$\max_{z\in\{0,1\}^{N\times K}}\; \sum_{k=1}^K \sum_{i<j,\,i,j\in F_k} \|x_i-x_j\|_2^2,$$

again with per-fold balance $|F_k| = s = N/K$ (Baumann et al., 9 Jan 2026).

2. Algorithmic Methodologies

Several algorithmic frameworks address the balanced $k$-fold assignment, motivated by different objectives:

a) Fixed-sized clusters $k$-Means:

• The assignment step reduces to a classical $n\times n$ linear sum assignment problem with costs $W_{a,i} = \|X_i - C_{\ell(a)}\|^2$, with slots $a$ grouped by fold and $\ell(a)$ mapping slots to folds.
• Solved using the Hungarian algorithm, which proceeds through row/column reduction, zero-covering, and optimal selection, with per-iteration complexity $O(n^3)$.
• The method is an exact alternating minimization: assignment (via Hungarian), then centroid update, until convergence (Malinen et al., 27 Jan 2025).

b) Assignment-Based Anticlustering (ABA):

• ABA maximizes intra-fold (within-group) diversity via a sequence of linear assignment problems, leveraging the equivalence

$$\sum_{i<j\in C}\|x_i-x_j\|^2 = n \sum_{i\in C}\|x_i-\mu\|^2$$

for centroids $\mu$.

• Data is sorted by squared distance from the global centroid and processed in batches of size $\approx K$, assigning each batch to one fold using LAPJV or Hungarian, and updating centroids incrementally.
• Hierarchical decomposition ($K=K_1K_2\ldots K_L$) controls subproblem size for scalability, yielding overall cost $O(NL K^{2/L})$ with $L$ typically 2 or 3 (Baumann et al., 9 Jan 2026).

c) Enumeration of Fold Configurations:

• For exact reproducibility and audit in $k$-fold cross-validation, all possible balanced multiclass fold matrices $F$ (with given class/fold marginals) can be systematically generated.
• Depth-first recursive algorithms enumerate nonnegative integer matrices $F$ obeying row/column sum constraints, with symmetry-breaking to account for fold label equivalence (Fazekas et al., 2024).

d) Pairwise Coverage via Balanced Incomplete Block Designs (BIBD):

• In applications like peer review, the goal is to assign $n$ objects to $r$ reviewers (“blocks”) of size $k$ so all pairs are covered, seeking $r$ minimized and near the lower bound $L(n,k)=\lceil n(n-1)/k(k-1)\rceil$.
• Classical constructions ([8]) and explicit BIBD constructions are compared, with new BIBD-based assignments achieving $r \leq 3/2\,L(n,k)$ in the regime $\sqrt{n}\leq k \leq n/2$ whenever $n/k$ is a prime power and $n$ divides $k^2$ (0909.3533).

3. Analytical Results and Combinatorial Properties

Balanced $k$-fold assignment manifests rich combinatorial structure:

• Enumeration of fold configurations: The number of standardized $k$-fold assignments (for multiclass contingency tables) is given by

$$Z_C(\mathbf{n},\mathbf{c}) = [t_1^{n_1}\cdots t_k^{n_k}] \prod_{j=1}^C (t_1+\cdots+t_k)^{c_j}$$

where $C$ is the number of classes, $\mathbf{c}$ the class sizes, and $\mathbf{n}$ the fold sizes. For binary class ($C=2$) and perfectly balanced $m=N/k$, there is an explicit inclusion-exclusion formula (Fazekas et al., 2024).

• Block design bounds: For pairwise covering, the lower bound $L(n,k)$ is achieved by a construction only when $k=\sqrt{n}$ and $k$ is a prime power; otherwise, optimal assignments are within $3/2$ of the lower bound for $\sqrt{n}\leq k\leq n$ (0909.3533).
• Assignment optimality: In fixed-sized clusters $k$-means, each assignment step is globally optimal for current centroids, and since the candidate assignments are finite, the method converges to a locally optimal balanced partition (Malinen et al., 27 Jan 2025).

4. Practical Algorithms and Scalability

Balanced $k$-fold assignment methods must address computational scalability:

Method	Complexity	Feasible $N$, $k$	Special Requirements
Fixed-sized $k$-means	$O(n^3)$ per iter.	$n\lesssim 5000$	Full $n\times n$ cost matrix, Hungarian algorithm (Malinen et al., 27 Jan 2025)
ABA	$O(N K^2)$ (base), $O(N L K^{2/L})$ (hierarchical)	$N\sim 10^6$, $K\sim 10^5$	LAPJV/auction solver, no $N\times N$ matrix, parallel
Exact configuration enumeration	$\sim$number of foldings $\times kC$	$N,C,k$ small (e.g. $N\lesssim 40$, $C,k\lesssim 10$)	Recursion, symmetry-breaking (Fazekas et al., 2024)
BIBD construction	NA (explicit math.)	$n/k$ prime power, $n|k^2$	Requires Latin squares, $k\geq \sqrt{n}$ (0909.3533)

ABA supports million-scale data and $K\sim 10^5$ without forming a full distance matrix; hierarchical splitting enables further scaling. In contrast, fixed-sized clusters $k$-means is bottlenecked by the cubic Hungarian step and is practical for $n\lesssim 5000$ (Baumann et al., 9 Jan 2026, Malinen et al., 27 Jan 2025).

5. Empirical Performance and Use Cases

Balanced clustering (fixed-sized $k$-means): Enables clustering of large datasets ($n>5000$) with specified cluster sizes and mean square error minimization (Malinen et al., 27 Jan 2025).

Anticlustering (ABA): For balanced $k$-fold cross-validation focused on diversity (i.e., representative folds), ABA yields higher intra-fold variance, improved objective ($0.05$–$0.2\%$ over METIS; $1$–$2\%$ over fast_anticlustering in large $K$), orders-of-magnitude faster runtime, and more uniform fold variance. Experiments on ImageNet32 ($n\sim 1.3\times 10^6$, $d=3072$, $K\leq 640,000$) run in under $8$ minutes, outperforming random and exchange-based methods (Baumann et al., 9 Jan 2026).

Contingency of reported cross-validation results: Exact enumeration of all balanced $k$-fold assignments allows ultimate consistency checks for claimed experimental setups, as implemented in open-source tools (Fazekas et al., 2024).

Peer review and covering designs: BIBD-based assignments efficiently minimize reviewer count for all-pair coverage, offering explicit constructions within $3/2$ of the information-theoretic minimum (0909.3533).

6. Implementation Considerations and Recommendations

• For moderate $k$ and $N$, fixed-sized $k$-means or ABA (base version) with LAPJV/Auction solvers can be directly deployed. For large $K$, hierarchical ABA presents a tractable path.
• ABA is deterministic once the initial ordering is fixed; randomized tie-breaking can be switched off for reproducibility (Baumann et al., 9 Jan 2026).
• Storing only $N\times d$ data and $K\times d$ centroids, ABA avoids the $O(N^2)$ storage burden.
• Exact enumeration should be used for small $N$, $C$, $k$ to guarantee exhaustiveness (Fazekas et al., 2024).
• For pairwise coverage, explicit BIBD constructions are available when parameter conditions permit, otherwise fallback to combinatorial bounds (0909.3533).

7. Theoretical and Applied Significance

Balanced $k$-fold assignment integrates core principles from assignment problems, integer programming, combinatorial design, and empirical machine learning methodology. It underpins principled cross-validation, balanced experimental design, stratified data partitioning, and robust peer review systems. Advances in scalable approximation algorithms (ABA), exact mathematical constructions (BIBD), and exhaustive combinatorial enumeration collectively support both large-scale empirical practice and foundational reproducibility in machine learning and data science (Malinen et al., 27 Jan 2025, Baumann et al., 9 Jan 2026, Fazekas et al., 2024, 0909.3533).