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 $k$0 or $k$1 points and $k$2.
• 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 $k$3 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$4-means clustering: Assignments

$k$5

subject to binary assignments $k$6 and fixed cluster sizes $k$7 (Malinen et al., 27 Jan 2025).

• Euclidean anticlustering (“max diversity”):

$k$8

again with per-fold balance $k$9 (Baumann et al., 9 Jan 2026).

2. Algorithmic Methodologies

Several algorithmic frameworks address the balanced $k$0-fold assignment, motivated by different objectives:

a) Fixed-sized clusters $k$1-Means:

• The assignment step reduces to a classical $k$2 linear sum assignment problem with costs $k$3, with slots $k$4 grouped by fold and $k$5 mapping slots to folds.
• Solved using the Hungarian algorithm, which proceeds through row/column reduction, zero-covering, and optimal selection, with per-iteration complexity $k$6.
• 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

$k$7

for centroids $k$8.

• Data is sorted by squared distance from the global centroid and processed in batches of size $k$9, assigning each batch to one fold using LAPJV or Hungarian, and updating centroids incrementally.
• Hierarchical decomposition ($k$0) controls subproblem size for scalability, yielding overall cost $k$1 with $k$2 typically 2 or 3 (Baumann et al., 9 Jan 2026).

c) Enumeration of Fold Configurations:

• For exact reproducibility and audit in $k$3-fold cross-validation, all possible balanced multiclass fold matrices $k$4 (with given class/fold marginals) can be systematically generated.
• Depth-first recursive algorithms enumerate nonnegative integer matrices $k$5 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 $k$6 objects to $k$7 reviewers (“blocks”) of size $k$8 so all pairs are covered, seeking $k$9 minimized and near the lower bound $k$0.
• Classical constructions ([8]) and explicit BIBD constructions are compared, with new BIBD-based assignments achieving $k$1 in the regime $k$2 whenever $k$3 is a prime power and $k$4 divides $k$5 (0909.3533).

3. Analytical Results and Combinatorial Properties

Balanced $k$6-fold assignment manifests rich combinatorial structure:

• Enumeration of fold configurations: The number of standardized $k$7-fold assignments (for multiclass contingency tables) is given by

$k$8

where $k$9 is the number of classes, $X=\{X_1,\dots,X_N\}$0 the class sizes, and $X=\{X_1,\dots,X_N\}$1 the fold sizes. For binary class ($X=\{X_1,\dots,X_N\}$2) and perfectly balanced $X=\{X_1,\dots,X_N\}$3, there is an explicit inclusion-exclusion formula (Fazekas et al., 2024).

• Block design bounds: For pairwise covering, the lower bound $X=\{X_1,\dots,X_N\}$4 is achieved by a construction only when $X=\{X_1,\dots,X_N\}$5 and $X=\{X_1,\dots,X_N\}$6 is a prime power; otherwise, optimal assignments are within $X=\{X_1,\dots,X_N\}$7 of the lower bound for $X=\{X_1,\dots,X_N\}$8 (0909.3533).
• Assignment optimality: In fixed-sized clusters $X=\{X_1,\dots,X_N\}$9-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 $\mathbb{R}^d$0-fold assignment methods must address computational scalability:

Method	Complexity	Feasible $\mathbb{R}^d$1, $\mathbb{R}^d$2	Special Requirements
Fixed-sized $\mathbb{R}^d$3-means	$\mathbb{R}^d$4 per iter.	$\mathbb{R}^d$5	Full $\mathbb{R}^d$6 cost matrix, Hungarian algorithm (Malinen et al., 27 Jan 2025)
ABA	$\mathbb{R}^d$7 (base), $\mathbb{R}^d$8 (hierarchical)	$\mathbb{R}^d$9, $k$0	LAPJV/auction solver, no $k$1 matrix, parallel
Exact configuration enumeration	$k$2number of foldings $k$3	$k$4 small (e.g. $k$5, $k$6)	Recursion, symmetry-breaking (Fazekas et al., 2024)
BIBD construction	NA (explicit math.)	$k$7 prime power, $k$8	Requires Latin squares, $k$9 (0909.3533)

ABA supports million-scale data and $F_1,\dots,F_k$0 without forming a full distance matrix; hierarchical splitting enables further scaling. In contrast, fixed-sized clusters $F_1,\dots,F_k$1-means is bottlenecked by the cubic Hungarian step and is practical for $F_1,\dots,F_k$2 (Baumann et al., 9 Jan 2026, Malinen et al., 27 Jan 2025).

5. Empirical Performance and Use Cases

Balanced clustering (fixed-sized $F_1,\dots,F_k$3-means): Enables clustering of large datasets ($F_1,\dots,F_k$4) with specified cluster sizes and mean square error minimization (Malinen et al., 27 Jan 2025).

Anticlustering (ABA): For balanced $F_1,\dots,F_k$5-fold cross-validation focused on diversity (i.e., representative folds), ABA yields higher intra-fold variance, improved objective ($F_1,\dots,F_k$6–$F_1,\dots,F_k$7 over METIS; $F_1,\dots,F_k$8–$F_1,\dots,F_k$9 over fast_anticlustering in large $F_i$0), orders-of-magnitude faster runtime, and more uniform fold variance. Experiments on ImageNet32 ($F_i$1, $F_i$2, $F_i$3) run in under $F_i$4 minutes, outperforming random and exchange-based methods (Baumann et al., 9 Jan 2026).

Contingency of reported cross-validation results: Exact enumeration of all balanced $F_i$5-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 $F_i$6 of the information-theoretic minimum (0909.3533).

6. Implementation Considerations and Recommendations

• For moderate $F_i$7 and $F_i$8, fixed-sized $F_i$9-means or ABA (base version) with LAPJV/Auction solvers can be directly deployed. For large $k$00, 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 $k$01 data and $k$02 centroids, ABA avoids the $k$03 storage burden.
• Exact enumeration should be used for small $k$04, $k$05, $k$06 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$07-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).