rank-based aggregation

This study addresses the limitations of traditional vector aggregation operators, which flatten matrices and thereby lose structural information while relying on decomposability. To overcome these issues, this work formally defines the theoretical framework of Matrix Aggregation Operators (MAOs) for the first time, systematically analyzing their decomposability and symmetry properties. Drawing upon fuzzy set theory and the maximum entropy principle, and integrating grouping functions with the MEOWA operator, it proposes the Maximum Entropy Global Covering Index (MEGCI) family along with its construction methodology. The research reveals the existence of non-decomposable operators and successfully constructs a series of MEGCI metrics. Computational experiments validate their effectiveness in clustering quality evaluation, thereby filling a critical theoretical gap in the field.

Traditional rank aggregation (RAP) produces only a single consensus ranking, failing to capture the diversity and inherent conflicts among input preferences. Method: This paper proposes the Optimal Set of Bucket Orders Problem (OSBOP), generalizing RAP from outputting a single bucket order (a weak ordering permitting ties) to generating a semantically complementary set of bucket orders. Input preferences are modeled via priority matrices, and the multi-solution consensus set is computed by integrating combinatorial optimization with heuristic search. Contribution/Results: OSBOP is the first RAP framework to adopt a set-based output while preserving interpretability and significantly improving fit quality. Experiments demonstrate that OSBOP substantially reduces the objective function value compared to the single-solution counterpart (OBOP); the resulting bucket orders exhibit clear functional specialization and mutual complementarity, jointly achieving high accuracy, diversity, and interpretability.

This work addresses the challenge of effectively aggregating heterogeneous probability distributions in multi-teacher knowledge distillation by proposing the first axiomatic framework for ensemble distillation. It formulates five core axioms that any valid knowledge aggregation operator should satisfy in the probability space and, leveraging operator theory and convex analysis, proves the existence—and non-uniqueness—of a family of operators fulfilling these axioms. The framework dispenses with the common assumption of teacher homogeneity, establishing generalization error and stability guarantees that are invariant to teacher heterogeneity. Theoretically, it reveals that multi-teacher aggregation simultaneously reduces both stochastic variance and systematic supervision bias, provides an upper bound on log-loss, ensures safe decay properties, and extends classical ensemble learning’s variance-reduction results to settings involving correlated errors.

This work addresses the challenge of formal reasoning and verification for Graph Neural Networks (GNNs). We propose Linear Inequality Modal Logic with Counting (LIMLₖ), the first modal logic featuring linear inequalities and counting modalities tailored to GNN semantics. LIMLₖ enables an efficient, bidirectional, and exact compilation between GNNs and logical formulas, precisely capturing the semantics of aggregation-combination GNNs and supporting explainability tasks such as GNN querying and equivalence checking. Theoretically, we prove that LIMLₖ satisfiability is PSPACE-complete—significantly extending the theoretical expressiveness frontier for GNN logics. Practically, LIMLₖ provides the first logical foundation for GNN formal verification that simultaneously ensures expressive completeness and computational feasibility.

This paper addresses ranking inconsistency in pairwise preference voting caused by cyclic dependencies. We propose a consensus ranking method based on *strong maximum circulation*: removing the strong maximum circulation from the directed vote graph yields an acyclic residual graph, from which a unique strong partial order—representing collective preference—is derived. We formally define strong maximum circulation and prove its existence and uniqueness; establish its duality with Kemeny ranking; and show that the minimum circulation removal problem is NP-hard and admits non-unique solutions. Integrating network flow optimization, linear programming duality, and circulation decomposition theory, our approach is solvable in polynomial time. It rigorously links circulation elimination to preference aggregation, providing an interpretable “collective tie” resolution mechanism for cyclic conflicts.

This work addresses the problem of weighted rank aggregation: efficiently merging multiple rankings over a common candidate set into a consensus ranking that minimizes the weighted distance to the input rankings. The authors propose the first unified approximation framework applicable to a variety of distance metrics—including Ulam, Spearman footrule, weighted Hamming, and Kendall-tau—by leveraging key structural properties to reduce the global 1-median optimization to small-scale local sorting subproblems. In the Massively Parallel Computation (MPC) model, their algorithm achieves a (2−ε)-approximation for all considered metrics using only a constant number of communication rounds, sublinear local memory per machine, and near-linear total memory in the number of candidates n. Notably, it attains a 1.968-approximation under the Ulam distance, improving upon prior results.

This study addresses how to aggregate individual preferences—each focused on distinct quantiles of outcome distributions, such as downside risk, typical performance, or upside potential—into collectively Pareto-efficient decisions. The authors develop a quantile preference model and employ an axiomatic approach combined with spectral weighting analysis to characterize social aggregation rules over general and elliptically contoured distribution domains. Their main contributions include a spectral support theorem showing that Pareto-consistent aggregation can only assign positive weight to quantiles already represented in society; an equivalence between representative-quantile aggregation and dictatorial mechanisms; and a full characterization of necessary and sufficient conditions for finite or threshold-based Pareto efficiency, along with the associated reducible structures.

This study addresses the problem of ranking and rank aggregation under matroid and flag-matroid prefix constraints, measured by Kendall tau distance. It unifies and generalizes existing notions such as k-fairness and block fairness, and for the first time handles more general constraints involving hierarchies and quotas. For the single-input setting, the authors propose a polynomial-time algorithm based on the Bruhat order and a greedy strategy to efficiently compute the nearest feasible ranking satisfying flag-matroid constraints. In the multi-input aggregation setting, they prove that the problem remains NP-hard even under partition matroids. By integrating matroid theory with structural analysis of the symmetric group, this work significantly extends the theoretical foundations of fair ranking.

This work addresses the tendency of standard rank aggregation methods to exacerbate underrepresentation of disadvantaged groups in applications such as hiring and recommendation. Focusing on fairness-aware rank aggregation under the Spearman footrule (i.e., L1 distance), the paper presents the first optimal algorithm for fair Top-$k$ ranking and improves the approximation ratio for fair full-ranking aggregation from 3 to 2. Through careful design of combinatorial optimization and approximation algorithms, the authors theoretically establish the optimality of their Top-$k$ solution and an improved approximation guarantee for the full-ranking setting. Extensive experiments on multiple real-world datasets demonstrate that the proposed methods significantly outperform existing baselines, effectively narrowing the performance gap between fair constrained rankings and their unconstrained counterparts.

This study addresses the problem of fairly breaking ties in ranking procedures under natural axiomatic constraints. By leveraging algebraic structures—specifically group actions, stabilizer subgroups, and orbit partitions—the authors develop an axiomatic framework to formally model tie-breaking mechanisms. Their main contributions include proving the incompatibility between anonymity and strict ranking rules, establishing that orbit partitioning is the unique tie-breaking method satisfying the proposed natural axioms, and introducing a general decomposition theorem that, for the first time, unifies and characterizes the intrinsic structure of various real-world ranking rules within a single theoretical framework.