compute worst-case ranking

This study investigates the robustness of subset rankings under ordinal aggregation by merging similar items in an item similarity graph, assuming additive evaluation metrics. The problem is formulated as four classes of combinatorial optimization tasks, aiming to maximize or minimize either the absolute or relative rank of a given subset. The work provides the first systematic characterization of the computational complexity of ranking optimization with partitioning operations, establishing NP-hardness for most variants while developing exact and approximation algorithms tailored to realistic, structured graph topologies. The proposed methodology is successfully applied to assess the robustness of rankings of greenhouse gas emission sources, demonstrating its practical utility across domains.

This paper addresses the computational inefficiency of differentiable permutation learning for large-scale, high-dimensional data. We propose a lightweight permutation modeling approach requiring only $N$ parameters—significantly fewer than Gumbel-Sinkhorn ($O(N^2)$) or low-rank approximations ($O(MN)$). Built upon the SoftSort continuous relaxation framework, our method decouples index reordering from differentiable sorting via iterative design, enabling compact and fully differentiable parameterization of the full permutation matrix. On tasks such as Self-Organizing Gaussians, memory consumption drops to $sim 1/N$ of conventional methods, while sorting accuracy substantially surpasses that of the original SoftSort. The approach exhibits strong scalability and practical deployability, bridging the gap between theoretical expressiveness and real-world efficiency in differentiable sorting and permutation learning.

Traditional optimality analyses rely solely on input size, ignoring inherent structural properties of problem instances. This limitation hinders fine-grained performance characterization and adaptive algorithm design. Method: We propose a generalized optimality paradigm that characterizes algorithmic efficiency via problem-relevant (including implicit) parameters, establishing a unified parametrized optimality framework. We formally define “universal optimality,” devise an adaptive analysis framework based on partitioned sorting, and introduce a novel, quantifiable metric for implicit orderliness. Contribution/Results: (1) We reinterpret the adaptivity boundaries of classical sorting algorithms under this refined lens; (2) we construct the first sorting algorithm provably achieving the information-theoretic lower bound with respect to our new metric; and (3) we provide a scalable, parametrized optimization paradigm applicable not only to sorting but also to broader algorithmic domains. This framework bridges instance-aware analysis and theoretical optimality, enabling more precise and structure-exploiting algorithm design.

This paper addresses NP-hard permutation problems—such as scheduling and graph ordering—by proposing an efficient approximation framework based on weak-order prediction. Methodologically, it pioneers the integration of Braverman–Mossel’s stochastic ranking theory with pairwise order prediction (i.e., predicting whether element *u* precedes *v*), showing that only a slight advantage over random guessing (accuracy ≥ 1/2 + ε) suffices to yield optimal or near-optimal permutations with high probability in polynomial time. The approach synergizes learning-augmented algorithm design, probabilistic analysis, and greedy/insertion-based refinement strategies, drastically reducing prediction query overhead. Crucially, this framework circumvents classical inapproximability barriers: unlike traditional worst-case approaches—whose runtime is exponentially lower-bounded without predictions—it enables rapid generation of high-quality solutions in time-sensitive applications, including real-time scheduling and network topology optimization.

This work addresses the challenge of revenue optimization in e-commerce search reranking, where maximizing platform revenue alone can degrade user experience or induce fraudulent behavior. To balance revenue with multidimensional constraints such as relevance, the problem is formulated as a constrained integer linear program (ILP). The authors propose PermR, a lightweight pairwise-swap heuristic algorithm that efficiently approximates optimal revenue while strictly satisfying constraints and meeting low-latency requirements. Offline experiments demonstrate that PermR achieves approximately 63% of the revenue gain attainable by the full ILP solution. Furthermore, a 14-day online A/B test involving 56 million queries shows a statistically significant 2% increase in revenue, confirming the method’s effectiveness and practicality in real-world deployment.

This work addresses the problem of achieving comparison-based sorting with near-optimal efficiency under the constraints of using only linear data moves and in-place operations. We propose a novel randomized in-place sorting algorithm that, for the first time, simultaneously achieves an expected $n \lg n + O(n)$ comparisons—matching the information-theoretic lower bound up to an additive linear term—and $O(n)$ data moves. The approach introduces a new ordered set structure supporting optimal searching and employs a parameterized design to balance worst-case overhead: for any integer $t$, it sorts using $n \lg n + O(n \lg^{(t)} n)$ comparisons and $O(tn)$ moves, significantly improving upon the previous best-known bound of $n \lg n + O(n \lg \lg n)$ comparisons.

This study investigates the parameterized complexity of clustering permutations under the Ulam metric, focusing on the k-center and k-median problems. Taking the number of centers \(k\) and the distance budget \(d\) as parameters, it establishes the first complete parameterized complexity landscape for these problems: Ulam k-center remains NP-hard even when \(d = 1\), yet is fixed-parameter tractable with respect to \(k + d\), albeit without a polynomial kernel; Ulam k-median is W[1]-hard with respect to \(d\), admits an XP algorithm, and possesses a polynomial kernel when parameterized by \(k + d\). The key technical contribution is a novel local search framework tailored to the non-local nature of Ulam distance, which also yields tight theoretical limits on kernelization possibilities.

This work investigates how to distinguish “easy” from “hard” input distributions in the stochastic caching problem to overcome the limitations imposed by worst-case analysis. To this end, we introduce subset entropy—a concept from information theory—as a novel parameter that enables the first fine-grained quantification of input distribution complexity. Building upon this measure, we develop a unified analytical framework applicable to both online and stochastic optimization settings. Within this framework, we establish competitive ratio upper bounds for classical algorithms such as LRU that explicitly depend on subset entropy. Our results demonstrate that under low-entropy distributions, these algorithms achieve substantially better performance than their classical worst-case guarantees, thereby providing new theoretical justification for the empirical effectiveness of LRU under realistic, structured inputs.

This study investigates the solvability and repair of permutation-matching puzzles on an $n \times n$ grid subject to row- and column-wise sorting constraints (either ascending or descending). By constructing a constraint graph, the work provides the first complete characterization of solvability conditions, introducing a “at most one switch” criterion to determine the existence of a solution. For solvable instances, it presents a counting method based on the hook-length formula; for unsolvable ones, it designs a linear-time algorithm to compute the minimum number of label flips required for repair. The framework is further extended to arbitrary permutation constraints, where the minimum repair problem is shown to be NP-complete. Integrating combinatorics, graph theory, and computational complexity, this work establishes a theoretical foundation and efficient algorithmic tools for this class of puzzles.