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The success of Constraint Programming relies partly on the global constraints and implementation of the associated filtering algorithms. Recently, new ideas emerged to improve these implementations in practice, especially regarding the all different constraint. In this paper, we consider the cardinality constraint with costs. The cardinality constraint is a generalization of the all different constraint that specifies the number of times each value must be taken by a given set of variables in a solution. The version with costs introduces an assignment cost and bounds the total sum of assignment costs. The arc consistency filtering algorithm of this constraint is difficult to use in practice, as it systematically searches for many shortest paths. We propose a new approach that works with upper bounds on shortest paths based on landmarks. This approach can be seen as a preprocessing. It is fast and avoids, in practice, a large number of explicit computations of shortest paths., Comment: Published at the 30th International Conference on Principles and Practice of Constraint Programming (CP 2024)

High-cardinality categorical features are a common characteristic of mixed-type tabular datasets. Existing generative model architectures struggle to learn the complexities of such data at scale, primarily due to the difficulty of parameterizing the categorical features. In this paper, we present a general variational autoencoder model, CardiCat, that can accurately fit imbalanced high-cardinality and heterogeneous tabular data. Our method substitutes one-hot encoding with regularized dual encoder-decoder embedding layers, which are jointly learned. This approach enables us to use embeddings that depend also on the other covariates, leading to a compact and homogenized parameterization of categorical features. Our model employs a considerably smaller trainable parameter space than competing methods, enabling learning at a large scale. CardiCat generates high-quality synthetic data that better represent high-cardinality and imbalanced features compared to competing VAE models for multiple real and simulated datasets.

We adopt a parametric approach to analyze the worst-case degradation in social welfare when the allocation of indivisible goods is constrained to be fair. Specifically, we are concerned with cardinality-constrained allocations, which require that each agent has at most $k$ items in their allocated bundle. We propose the notion of the price of cardinality, which captures the worst-case multiplicative loss of utilitarian or egalitarian social welfare resulting from imposing the cardinality constraint. We then characterize tight or almost-tight bounds on the price of cardinality as exact functions of the instance parameters, demonstrating how the social welfare improves as $k$ is increased. In particular, one of our main results refines and generalizes the existing asymptotic bound on the price of balancedness, as studied by Bei et al. [BLMS21]. We also further extend our analysis to the problem where the items are partitioned into disjoint categories, and each category has its own cardinality constraint. Through a parametric study of the price of cardinality, we provide a framework which aids decision makers in choosing an ideal level of cardinality-based fairness, using their knowledge of the potential loss of utilitarian and egalitarian social welfare., Comment: To appear in the proceeding of AAAI2025

Index tracking is a popular passive investment strategy aimed at optimizing portfolios, but fully replicating an index can lead to high transaction costs. To address this, partial replication have been proposed. However, the cardinality constraint renders the problem non-convex, non-differentiable, and often NP-hard, leading to the use of heuristic or neural network-based methods, which can be non-interpretable or have NP-hard complexity. To overcome these limitations, we propose a Differentiable Cardinality Constraint ($\textbf{DCC}$) for index tracking and introduce a floating-point precision-aware method ($\textbf{DCC}_{fpp}$) to address implementation issues. We theoretically prove our methods calculate cardinality accurately and enforce actual cardinality with polynomial time complexity. We propose the range of the hyperparameter $a$ ensures that $\textbf{DCC}_{fpp}$ has no error in real implementations, based on theoretical proof and experiment. Our method applied to mathematical method outperforms baseline methods across various datasets, demonstrating the effectiveness of the identified hyperparameter $a$., Comment: 10 pages, 6 figures, AAAI 2025 (accepted, but not published)

Combinatorial bilevel congestion pricing (CBCP), a variant of the discrete network design problem, seeks to minimize the total travel time experienced by all travelers in a road network, by strategically selecting toll locations and determining the corresponding charges. Conventional wisdom suggests that these problems are intractable since they have to be formulated and solved with a significant number of integer variables. Here, we devise a scalable local algorithm for the CBCP problem that guarantees convergence to a Kuhn-Tucker-Karush point. Our approach is novel in that it eliminates the use of integer variables altogether, instead introducing a cardinality constraint that limits the number of toll locations to a user-specified upper bound. The resulting bilevel program with the cardinality constraint is then transformed into a block-separable, single-level optimization problem that can be solved efficiently after penalization and decomposition. We are able to apply the algorithm to solve, in about 20 minutes, a CBCP instance with up to 3,000 links, of which hundreds can be tolled. To the best of our knowledge, no existing algorithm can solve CBCP problems at such a scale while providing any assurance of convergence.

Query-driven learned estimators are accurate, flexible, and lightweight alternatives to traditional estimators in query optimization. However, existing query-driven approaches struggle with the Out-of-distribution (OOD) problem, where the test workload distribution differs from the training workload, leading to performancedegradation. In this paper, we present CardOOD, a general learning framework designed to construct robust query-driven cardinality estimators that are resilient against the OOD problem. Our framework focuses on offline training algorithms that develop one-off models from a static workload, suitable for model initialization and periodic retraining. In CardOOD, we extend classical transfer/robust learning techniques to train query-driven cardinalityestimators, and the algorithms fall into three categories: representation learning, data manipulation, and new learning strategies. As these learning techniques are originally evaluated in computervision tasks, we also propose a new learning algorithm that exploits the property of cardinality estimation. This algorithm, lying in the category of new learning strategy, models the partial order constraint of cardinalities by a self-supervised learning task. Comprehensive experimental studies demonstrate the efficacy of the algorithms of CardOOD in mitigating the OOD problem to varying extents. We further integrate CardOOD into PostgreSQL, showcasing its practical utility in query optimization.

This paper presents a novel proof that for any convex cone, the size of conically independent generators is at most twice that of minimum cardinality generators. While this result is known for linear spaces, we extend it to general cones through a decomposition into linear and pointed components. Our constructive approach leads to a polynomial-time algorithm for computing minimum cardinality generators of finitely generated cones, improving upon existing methods that only compute conically independent generators., Comment: 6 pages

Private set intersection (PSI) and private set union (PSU) are the crucial primitives in secure multiparty computation protocols, which enable several participants to jointly compute the intersection and union of their private sets without revealing any additional information. Quantum homomorphic encryption (QHE) offers significant advantages in handling privacy-preserving computations. However, given the current limitations of quantum resources, developing efficient and feasible QHE-based protocols for PSI and PSU computations remains a critical challenge. In this work, a novel quantum private set intersection and union cardinality protocol is proposed, accompanied by the corresponding quantum circuits. Based on quantum homomorphic encryption, the protocol allows the intersection and union cardinality of users' private sets to be computed on quantum-encrypted data with the assistance of a semi-honest third party. By operating on encrypted quantum states, it effectively mitigates the risk of original information leakage. Furthermore, the protocol requires only simple Pauli and CNOT operations, avoiding the use of complex quantum manipulations (e.g., $T$ gate and phase rotation gate). Compared to related protocols, this approach offers advantages in feasibility and privacy protection., Comment: 15 pages, 6 figures

High\-cardinality categorical variables pose significant challenges in machine learning, particularly in terms of computational efficiency and model interpretability. Traditional one\-hot encoding often results in high\-dimensional sparse feature spaces, increasing the risk of overfitting and reducing scalability. This paper introduces novel encoding techniques, including means encoding, low\-rank encoding, and multinomial logistic regression encoding, to address these challenges. These methods leverage sufficient representations to generate compact and informative embeddings of categorical data. We conduct rigorous theoretical analyses and empirical validations on diverse datasets, demonstrating significant improvements in model performance and computational efficiency compared to baseline methods. The proposed techniques are particularly effective in domains requiring scalable solutions for large datasets, paving the way for more robust and efficient applications in machine learning., Comment: 2025 International Conference on Advanced Machine Learning and Data Science (AMLDS 2025)

This article presents nonlinear conjugate gradient methods for finding local weakly minimal points of set-valued optimization problems under a lower set less ordering relation. The set-valued objective function of the optimization problem under consideration is defined by finitely many continuously differentiable vector-valued functions. For such optimization problems, at first, we propose a general scheme for nonlinear conjugate gradient methods and then introduce Dai-Yuan, Polak-Ribi{\`e}re-Polyak, and Hestenes-Stiefel conjugate gradient parameters for set-valued functions. Toward deriving the general scheme, we introduce a condition of sufficient decrease and Wolfe line searches for set-valued functions. For a given sequence of descent directions of a set-valued function, it is found that if the proposed standard Wolfe line search technique is employed, then the generated sequence of iterates for set optimization follows a Zoutendijk-like condition. With the help of the derived Zoutendijk-like condition, we report that all the proposed nonlinear conjugate gradient schemes are globally convergent under usual assumptions. It is important to note that the ordering cone used in the entire study is not restricted to be finitely generated, and no regularity assumption on the solution set of the problem is required for any of the reported convergence analyses. Finally, we demonstrate the performance of the proposed methods through numerical experiments. In the numerical experiments, we demonstrate the effectiveness of the proposed methods not only on the commonly used test instances for set optimization but also on a few newly introduced problems under general ordering cones that are neither nonnegative hyper-octant nor finitely generated.

If we treat the symmetric group $S_n$ as a probability measure space where each element has measure $1/n!$, then the number of cycles in a permutation becomes a random variable. The Cycle Length Lemma describes the expected values of products of these random variables. Here we categorify the Cycle Length Lemma by showing that it follows from an equivalence between groupoids., Comment: 6 pages

Recent works have explored the use of counting queries coupled with Description Logic ontologies. The answer to such a query in a model of a knowledge base is either an integer or $\infty$, and its spectrum is the set of its answers over all models. While it is unclear how to compute and manipulate such a set in general, we identify a class of counting queries whose spectra can be effectively represented. Focusing on atomic counting queries, we pinpoint the possible shapes of a spectrum over $\mathcal{ALCIF}$ ontologies: they are essentially the subsets of $\mathbb{N} \cup \{ \infty \}$ closed under addition. For most sublogics of $\mathcal{ALCIF}$, we show that possible spectra enjoy simpler shapes, being $[ m, \infty ]$ or variations thereof. To obtain our results, we refine constructions used for finite model reasoning and notably rely on a cycle-reversion technique for the Horn fragment of $\mathcal{ALCIF}$. We also study the data complexity of computing the proposed effective representation and establish the $\mathsf{FP}^{\mathsf{NP}[\log]}$-completeness of this task under several settings., Comment: 26 pages

Cardinality Estimation is to estimate the size of the output of a query without computing it, by using only statistics on the input relations. Existing estimators try to return an unbiased estimate of the cardinality: this is notoriously difficult. A new class of estimators have been proposed recently, called "pessimistic estimators", which compute a guaranteed upper bound on the query output. Two recent advances have made pessimistic estimators practical. The first is the recent observation that degree sequences of the input relations can be used to compute query upper bounds. The second is a long line of theoretical results that have developed the use of information theoretic inequalities for query upper bounds. This paper is a short overview of pessimistic cardinality estimators, contrasting them with traditional estimators.

In this article, we investigate the cardinality of Grobner basis under various orderings. We identify a family of polynomials F and a criteria for the monomial orderings such that the reduced Grobner basis is double exponential in cardinality. We also show that the said criteria is satisfied by orderings such as lexicographic, degree lexicographic and weighted orderings., Comment: 18 pages,2 figures

Cardinality sketches are compact data structures for representing sets or vectors, enabling efficient approximation of their cardinality (or the number of nonzero entries). These sketches are space-efficient, typically requiring only logarithmic storage relative to input size, and support incremental updates, allowing for dynamic modifications. A critical property of many cardinality sketches is composability, meaning that the sketch of a union of sets can be computed from individual sketches. Existing designs typically provide strong statistical guarantees, accurately answering an exponential number of queries in terms of sketch size $k$. However, these guarantees degrade to quadratic in $k$ when queries are adaptive and may depend on previous responses. Prior works on statistical queries (Steinke and Ullman, 2015) and specific MinHash cardinality sketches (Ahmadian and Cohen, 2024) established that the quadratic bound on the number of adaptive queries is, in fact, unavoidable. In this work, we develop a unified framework that generalizes these results across broad classes of cardinality sketches. We show that any union-composable sketching map is vulnerable to attack with $\tilde{O}(k^4)$ queries and, if the sketching map is also monotone (as for MinHash and statistical queries), we obtain a tight bound of $\tilde{O}(k^2)$ queries. Additionally, we demonstrate that linear sketches over the reals $\mathbb{R}$ and fields $\mathbb{F}_p$ can be attacked using $\tilde{O}(k^2)$ adaptive queries, which is optimal and strengthens some of the recent results by Gribelyuk et al. (2024), which required a larger polynomial number of rounds for such matrices.

We consider two classic problems: maximum coverage and monotone submodular maximization subject to a cardinality constraint. [Nemhauser--Wolsey--Fisher '78] proved that the greedy algorithm provides an approximation of $1-1/e$ for both problems, and it is known that this guarantee is tight ([Nemhauser--Wolsey '78; Feige '98]). Thus, one would naturally assume that everything is resolved when considering the approximation guarantees of these two problems, as both exhibit the same tight approximation and hardness. In this work we show that this is not the case, and study both problems when the cardinality constraint is a constant fraction $c \in (0,1]$ of the ground set. We prove that monotone submodular maximization subject to a cardinality constraint admits an approximation of $1-(1-c)^{1/c}$; This approximation equals $1$ when $c=1$ and it gracefully degrades to $1-1/e$ when $c$ approaches $0$. Moreover, for every $c=1/s$ (for any integer $s \in \mathbb{N}$) we present a matching hardness. Surprisingly, for $c=1/2$ we prove that Maximum Coverage admits an approximation of $0.7533$, thus separating the two problems. To the best of our knowledge, this is the first known example of a well-studied maximization problem for which coverage and monotone submodular objectives exhibit a different best possible approximation., Comment: 17 pages

We propose an advancement in cardinality estimation by augmenting autoregressive models with a traditional grid structure. The novel hybrid estimator addresses the limitations of autoregressive models by creating a smaller representation of continuous columns and by incorporating a batch execution for queries with range predicates, as opposed to an iterative sampling approach. The suggested modification markedly improves the execution time of the model for both training and prediction, reduces memory consumption, and does so with minimal decline in accuracy. We further present an algorithm that enables the estimator to calculate cardinality estimates for range join queries efficiently. To validate the effectiveness of our cardinality estimator, we conduct and present a comprehensive evaluation considering state-of-the-art competitors using three benchmark datasets -- demonstrating vast improvements in execution times and resource utilization., Comment: 13 pages, 6 figures, 9 tables

Cardinality estimation (CE) plays a crucial role in many database-related tasks such as query generation, cost estimation, and join ordering. Lately, we have witnessed the emergence of numerous learned CE models. However, no single CE model is invincible when it comes to the datasets with various data distributions. To facilitate data-intensive applications with accurate and efficient cardinality estimation, it is important to have an approach that can judiciously and efficiently select the most suitable CE model for an arbitrary dataset. In this paper, we study a new problem of selecting the best CE models for a variety of datasets. This problem is rather challenging as it is hard to capture the relationship from various datasets to the performance of disparate models. To address this problem, we propose a model advisor, named AutoCE, which can adaptively select the best model for a dataset. The main contribution of AutoCE is the learning-based model selection, where deep metric learning is used to learn a recommendation model and incremental learning is proposed to reduce the labeling overhead and improve the model robustness. We have integrated AutoCE into PostgreSQL and evaluated its impact on query optimization. The results showed that AutoCE achieved the best performance (27% better) and outperformed the baselines concerning accuracy (2.1 times better) and efficacy (4.2 times better).

Cardinality estimation (CE) plays a crucial role in database optimizer. We have witnessed the emergence of numerous learned CE models recently which can outperform traditional methods such as histograms and samplings. However, learned models also bring many security risks. For example, a query-driven learned CE model learns a query-to-cardinality mapping based on the historical workload. Such a learned model could be attacked by poisoning queries, which are crafted by malicious attackers and woven into the historical workload, leading to performance degradation of CE. In this paper, we explore the potential security risks in learned CE and study a new problem of poisoning attacks on learned CE in a black-box setting. Experiments show that PACE reduces the accuracy of the learned CE models by 178 times, leading to a 10 times decrease in the end-to-end performance of the target database.

It is known since 1975 (\cite{HK75}) that maximum cardinality matchings in bipartite graphs with $n$ nodes and $m$ edges can be computed in time $O(\sqrt{n} m)$. Asymptotically faster algorithms were found in the last decade and maximum cardinality bipartite matchings can now be computed in near-linear time~\cite{NearlyLinearTimeBipartiteMatching, AlmostLinearTimeMaxFlow,AlmostLinearTimeMinCostFlow}. For general graphs, the problem seems harder. Algorithms with running time $O(\sqrt{n} m)$ were given in~\cite{MV80,Vazirani94,Vazirani12,Vazirani20,Vazirani23,Goldberg-Karzanov,GT91,Gabow:GeneralMatching}. Mattingly and Ritchey~\cite{Mattingly-Ritchey} and Huang and Stein~\cite{Huang-Stein} discuss implementations of the Micali-Vazirani Algorithm. We describe an implementation of Gabow's algorithm~\cite{Gabow:GeneralMatching} in C++ based on LEDA~\cite{LEDAsystem,LEDAbook} and report on running time experiments. On worst-case graphs, the asymptotic improvement pays off dramatically. On random graphs, there is no improvement with respect to algorithms that have a worst-case running time of $O(n m)$. The performance seems to be near-linear. The implementation is available open-source.

Significant work has been done on computing the ``average'' optimal solution value for various $\mathsf{NP}$-complete problems using the Erd\"{o}s-R\'{e}nyi model to establish \emph{critical thresholds}. Critical thresholds define narrow bounds for the optimal solution of a problem instance such that the probability that the solution value lies outside these bounds vanishes as the instance size approaches infinity. In this paper, we extend the Erd\"{o}s-R\'{e}nyi model to general hypergraphs on $n$ vertices and $M$ hyperedges. We consider the problem of determining critical thresholds for the largest cardinality matching, and we show that for $M=o(1.155^n)$ the size of the maximum cardinality matching is almost surely 1. On the other hand, if $M=\Theta(2^n)$ then the size of the maximum cardinality matching is $\Omega(n^{\frac12-\gamma})$ for an arbitrary $\gamma >0$. Lastly, we address the gap where $\Omega(1.155^n)=M=o(2^n)$ empirically through computer simulations.

In hypergraphs an edge that crosses a cut can be split in several ways, depending on how many nodes are placed on each side of the cut. A cardinality-based splitting function assigns a nonnegative cost of $w_i$ for each cut hyperedge $e$ with exactly $i$ nodes on the side of the cut that contains the minority of nodes from $e$. The cardinality-based minimum $s$-$t$ cut aims to find an $s$-$t$ cut with minimum total cost. Assuming the costs $w_i$ are polynomially bounded by the input size and $w_0=0$ and $w_1=1$, we show that the problem becomes NP-hard outside the submodular region found by Veldt et al. Our result also holds for $k$-uniform hypergraphs with $k \geq 4$. Specifically for $4$-uniform hypergraphs we show that the problem is NP-hard for all $w_2>2$, and additionally prove that the \textsc{No-Even-Split} problem is NP-hard.

An important classical result in ZFC asserts that every infinite cardinal number is idempotent. Using this fact, we obtain several algebraic results in this article. The first result asserts that an infinite Abelian group has a proper subgroup with the same cardinality if and only if it is not a Pr\"ufer group. In the second result, the cardinality of any monoid-ring $R[M]$ (not necessarily commutative) is calculated. In particular, the cardinality of every polynomial ring with any number of variables (possibly infinite) is easily computed. Next, it is shown that every commutative ring and its total ring of fractions have the same cardinality. This set-theoretic observation leads us to a notion in ring theory that we call a balanced ring (i.e. a ring that is canonically isomorphic to its total ring of fractions). Every zero-dimensional ring is a balanced ring. Then we show that a Noetherian ring is a balanced ring if and only if its localization at every maximal ideal has zero depth. It is also proved that every self-injective ring (injective as a module over itself) is a balanced ring., Comment: 9 pages

Let $A\subseteq \mathbb{Z}_p^2$ be a set of size $2p+1$ for prime $p\geq 5$. In this paper, we prove that $A\hat{+}A=\{a_1+a_2\mid a_1,a_2\in A, a_1\neq a_2\}$ has cardinality at least $4p$. This result is the first advancement in over two decades on a variant of the Erd\H{o}s-Heilbronn problem studied by Eliahou and Kervaire., Comment: 20 pages, preprint to be submitted shortly Edit 1: Added email and fixed some typos. Edit 2: Fixed more typos. Thank you Zhiyu Yuan for finding these

A minimum $s$-$t$ cut in a hypergraph is a bipartition of vertices that separates two nodes $s$ and $t$ while minimizing a hypergraph cut function. The cardinality-based hypergraph cut function assigns a cut penalty to each hyperedge based on the number of nodes in the hyperedge that are on each side of the split. Previous work has shown that when hyperedge cut penalties are submodular, this problem can be reduced to a graph $s$-$t$ cut problem and hence solved in polynomial time. NP-hardness results are also known for a certain class of non-submodular penalties, though the complexity remained open in many parameter regimes. In this paper we highlight and leverage a connection to Valued Constraint Satisfaction Problems to show that the problem is NP-hard for all non-submodular hyperedge cut penalty, except for one trivial case where a 0-cost solution is always possible. We then turn our attention to approximation strategies and approximation hardness results in the non-submodular case. We design a strategy for projecting non-submodular penalties to the submodular region, which we prove gives the optimal approximation among all such projection strategies. We also show that alternative approaches are unlikely to provide improved guarantees, by showing it is UGC-hard to obtain a better approximation in the simplest setting where all hyperedges have exactly 4 nodes.

Hypergraphs provide a robust framework for modeling complex systems with higher-order interactions. However, analyzing them in dynamic settings presents significant computational challenges. To address this, we introduce a novel method that adapts the cardinality-based gadget to convert hypergraphs into strongly connected weighted directed graphs, complemented by a symmetrized combinatorial Laplacian. We demonstrate that the harmonic mean of the conductance and edge expansion of the original hypergraph can be upper-bounded by the conductance of the transformed directed graph, effectively preserving crucial cut information. Additionally, we analyze how the resulting Laplacian relates to that derived from the star expansion. Our approach was validated through change point detection experiments on both synthetic and real datasets, showing superior performance over clique and star expansions in maintaining spectral information in dynamic settings. Finally, we applied our method to analyze a dynamic legal hypergraph constructed from extensive United States court opinion data.

A subset of $\mathbb{F}_q^2$ is called an arc if it does not contain three collinear points. We show that there are at most $\binom{(1 + o(1))q}{m}$ arcs of size $m \gg q^{1/2} (\log q)^{3/2}$, nearly matching a trivial lower bound $\binom{q}{m}$. This was previously known to hold for $m \gg q^{2/3} (\log q)^3$, due to Bhowmick and Roche-Newton. The lower bound on $m$ is best possible up to a logarithmic factor., Comment: 3 pages

This paper introduces a novel approach to interactive robots by leveraging the form-factor of cards to create thin robots equipped with vibrational capabilities for locomotion and haptic feedback. The system is composed of flat-shaped robots with on-device sensing and wireless control, which offer lightweight portability and scalability. This research introduces a hardware prototype. Applications include augmented card playing, educational tools, and assistive technology, which showcase CARDinality's versatility in tangible interaction., Comment: Accepted for ACM UIST 2024

Modern Cardinality Estimators struggle with data updates. This research tackles this challenge within single-table. We introduce ICE, an Index-based Cardinality Estimator, the first data-driven estimator that enables instant, tuple-leveled updates. ICE has learned two key lessons from the multidimensional index and applied them to solve cardinality estimation in dynamic scenarios: (1) Index possesses the capability for swift training and seamless updating amidst vast multidimensional data. (2) Index offers precise data distribution, staying synchronized with the latest database version. These insights endow the index with the ability to be a highly accurate, data-driven model that rapidly adapts to data updates and is resilient to out-of-distribution challenges during query testing. To make a solitary index support cardinality estimation, we have crafted sophisticated algorithms for training, updating, and estimating, analyzing unbiasedness and variance. Extensive experiments demonstrate the superiority of ICE. ICE offers precise estimations and fast updates/construction across diverse workloads. Compared to state-of-the-art real-time query-driven models, ICE boasts superior accuracy (2-3 orders of magnitude more precise), faster updates (4.7-6.9 times faster), and significantly reduced training time (up to 1-3 orders of magnitude faster).

Cardinality estimation is crucial for enabling high query performance in relational databases. Recently learned cardinality estimation models have been proposed to improve accuracy but there is no systematic benchmark or datasets which allows researchers to evaluate the progress made by new learned approaches and even systematically develop new learned approaches. In this paper, we are releasing a benchmark, containing thousands of queries over 20 distinct real-world databases for learned cardinality estimation. In contrast to other initial benchmarks, our benchmark is much more diverse and can be used for training and testing learned models systematically. Using this benchmark, we explored whether learned cardinality estimation can be transferred to an unseen dataset in a zero-shot manner. We trained GNN-based and transformer-based models to study the problem in three setups: 1-) instance-based, 2-) zero-shot, and 3-) fine-tuned. Our results show that while we get promising results for zero-shot cardinality estimation on simple single table queries; as soon as we add joins, the accuracy drops. However, we show that with fine-tuning, we can still utilize pre-trained models for cardinality estimation, significantly reducing training overheads compared to instance specific models. We are open sourcing our scripts to collect statistics, generate queries and training datasets to foster more extensive research, also from the ML community on the important problem of cardinality estimation and in particular improve on recent directions such as pre-trained cardinality estimation.

Functional dependencies (FDs) are a central theme in databases, playing a major role in the design of database schemas and the optimization of queries. In this work, we introduce the {\it targeted least cardinality candidate key problem} (TCAND). This problem is defined over a set of functional dependencies $F$ and a target variable set $T \subseteq V$, and it aims to find the smallest set $X \subseteq V$ such that the FD $X \to T$ can be derived from $F$. The TCAND problem generalizes the well-known NP-hard problem of finding the least cardinality candidate key~\cite{lucchesi1978candidate}, which has been previously demonstrated to be at least as difficult as the set cover problem. We present an integer programming (IP) formulation for the TCAND problem, analogous to a layered set cover problem. We analyze its linear programming (LP) relaxation from two perspectives: we propose two approximation algorithms and investigate the integrality gap. Our findings indicate that the approximation upper bounds for our algorithms are not significantly improvable through LP rounding, a notable distinction from the standard set cover problem. Additionally, we discover that a generalization of the TCAND problem is equivalent to a variant of the set cover problem, named red-blue set cover~\cite{carr1999red}, which cannot be approximated within a sub-polynomial factor in polynomial time under plausible conjectures~\cite{chlamtavc2023approximating}. Despite the extensive history surrounding the issue of identifying the least cardinality candidate key, our research contributes new theoretical insights, novel algorithms, and demonstrates that the general TCAND problem poses complexities beyond those encountered in the set cover problem.

In the context of satellite monitoring of the earth, we can assume that the surface of the earth is divided into a set of regions. We assume that the impact of a big social/environmental event spills into neighboring regions. Using Identifying Code Sets (ICSes), we can deploy sensors in such a way that the region in which an event takes place can be uniquely identified, even with fewer sensors than regions. As Earth is almost a sphere, we use a soccer ball as a model. We construct a Soccer Ball Graph (SBG), and provide human-oriented, analytical proofs that 1) the SBG has at least 26 ICSes of cardinality ten, implying that there are at least 26 different ways to deploy ten satellites to monitor the Earth and 2) that the cardinality of the minimum Identifying Code Set (MICS) for the SBG is at least nine. We then provide a machine-oriented formal proof that the cardinality of the MICS for the SBG is in fact ten, meaning that one must deploy at least ten satellites to monitor the Earth in the SBG model. We also provide machine-oriented proof that there are exactly 26 ICSes of cardinality ten for the SBG., Comment: 22 pages, 5 figures, preprint

We present a detailed study of cardinality-aware top-$k$ classification, a novel approach that aims to learn an accurate top-$k$ set predictor while maintaining a low cardinality. We introduce a new target loss function tailored to this setting that accounts for both the classification error and the cardinality of the set predicted. To optimize this loss function, we propose two families of surrogate losses: cost-sensitive comp-sum losses and cost-sensitive constrained losses. Minimizing these loss functions leads to new cardinality-aware algorithms that we describe in detail in the case of both top-$k$ and threshold-based classifiers. We establish $H$-consistency bounds for our cardinality-aware surrogate loss functions, thereby providing a strong theoretical foundation for our algorithms. We report the results of extensive experiments on CIFAR-10, CIFAR-100, ImageNet, and SVHN datasets demonstrating the effectiveness and benefits of our cardinality-aware algorithms., Comment: arXiv admin note: substantial text overlap with arXiv:2403.19625

Cardinality-based feature models permit to select multiple copies of the same feature, thus generalizing the notion of product configurations from subsets of Boolean features to multisets of feature instances. This increased expressiveness shapes a-priori infinite and non-convex configuration spaces, which renders established solution-space mappings based on Boolean presence conditions insufficient for cardinality-based feature models. To address this issue, we propose weighted automata over featured multiset semirings as a novel behavioral variability modeling formalism for cardinality-based feature models. The formalism uses multisets over features as a predefined semantic domain for transition weights. It permits to use any algebraic structure forming a proper semiring on multisets to aggregate the weights traversed along paths to map accepted words to multiset configurations. In particular, tropical semirings constitute a promising sub-class with a reasonable trade-off between expressiveness and computational tractability of canonical analysis problems. The formalism is strictly more expressive than featured transition systems, as it enables upper-bound multiplicity constraints depending on the length of words. We provide a tool implementation of the behavioral variability model and present preliminary experimental results showing applicability and computational feasibility of the proposed approach., Comment: This is the author's version of the work. The definitive version will be published in Proceedings of 28th ACM International Systems and Software Product Lines Conference (SPLC'24)

For every prime number p and integer $n>1$, a simple, involutive, non-degenerate set-theoretic solution $(X,r$) of the Yang-Baxter equation of cardinality $|X| = p^n$ is constructed. Furthermore, for every non-(square-free) positive integer m which is not the square of a prime number, a non-simple, indecomposable, irretractable, involutive, non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation of cardinality $|X| = m$ is constructed. A recent question of Castelli on the existence of singular solutions of certain type is also answered affirmatively., Comment: arXiv admin note: text overlap with arXiv:2401.12904, arXiv:2212.06753, arXiv:2112.07271, arXiv:2012.08400

Estimating cardinality, i.e., the number of distinct elements, of a data stream is a fundamental problem in areas like databases, computer networks, and information retrieval. This study delves into a broader scenario where each element carries a positive weight. Unlike traditional cardinality estimation, limited research exists on weighted cardinality, with current methods requiring substantial memory and computational resources, challenging for devices with limited capabilities and real-time applications like anomaly detection. To address these issues, we propose QSketch, a memory-efficient sketch method for estimating weighted cardinality in streams. QSketch uses a quantization technique to condense continuous variables into a compact set of integer variables, with each variable requiring only 8 bits, making it 8 times smaller than previous methods. Furthermore, we leverage dynamic properties during QSketch generation to significantly enhance estimation accuracy and achieve a lower time complexity of $O(1)$ for updating estimations upon encountering a new element. Experimental results on synthetic and real-world datasets show that QSketch is approximately 30\% more accurate and two orders of magnitude faster than the state-of-the-art, using only $1/8$ of the memory., Comment: 12 pages, 10 figures, accepted by KDD 2024

Cardinality estimation (CardEst) is essential for optimizing query execution plans. Recent ML-based CardEst methods achieve high accuracy but face deployment challenges due to high preparation costs and lack of transferability across databases. In this paper, we propose PRICE, a PRetrained multI-table CardEst model, which addresses these limitations. PRICE takes low-level but transferable features w.r.t. data distributions and query information and elegantly applies self-attention models to learn meta-knowledge to compute cardinality in any database. It is generally applicable to any unseen new database to attain high estimation accuracy, while its preparation cost is as little as the basic one-dimensional histogram-based CardEst methods. Moreover, PRICE can be finetuned to further enhance its performance on any specific database. We pretrained PRICE using 30 diverse datasets, completing the process in about 5 hours with a resulting model size of only about 40MB. Evaluations show that PRICE consistently outperforms existing methods, achieving the highest estimation accuracy on several unseen databases and generating faster execution plans with lower overhead. After finetuning with a small volume of databasespecific queries, PRICE could even find plans very close to the optimal ones. Meanwhile, PRICE is generally applicable to different settings such as data updates, data scaling, and query workload shifts. We have made all of our data and codes publicly available at https://github.com/StCarmen/PRICE.

For a dust-like self-similar set (generated by IFSs with the strong separation condition), Elekes, Keleti and M\'{a}th\'{e} found an invariant, called `algebraic dependence number', by considering its generating IFSs and isometry invariant self-similar measures. We find an intrinsic quantitative characterisation of this number: it is the dimension over $\mathbb{Q}$ of the vector space generated by the logarithms of all the common ratios of infinite geometric sequences in the gap length set, minus 1. With this concept, we present a lower bound on the cardinality of generating IFS (with or without separation conditions) in terms of the gap lengths of a dust-like set. We also establish analogous result for dust-like graph-directed attractors on complete metric spaces. This is a new application of the ratio analysis method and the gap sequence.

A group is Artinian if there is no infinite strictly descending chain of subgroups. Ol'shanskii has asked whether there are Artinian groups of arbitrarily large cardinality. We show that this problem is essentially the same as an analogous question, regarding universal algebras, asked by J\'onsson in the 1960s. We further show that these problems are the same as the so-called free subset problem. As a result, one can have a consistent strong negative answer (from a large cardinal assumption) as well as a consistent positive answer.

A perfect strategy is one that allows the mutually in-communicated players of a nonlocal game to win every trial of the game. Perfect strategies are basic tools for some fundamental results in quantum computation and crucial resources for some applications in quantum information. Here, we address the problem of producing qudit-qudit perfect quantum strategies with a small number of settings. For that, we exploit a recent result showing that any perfect quantum strategy induces a Kochen-Specker set. We identify a family of KS sets in even dimension $d \ge 6$ that, for many dimensions, require the smallest number of orthogonal bases known: $d+1$. This family was only defined for some $d$. We first extend the family to infinitely many more dimensions. Then, we show the optimal way to use each of these sets to produce a bipartite perfect strategy with minimum input cardinality. As a result, we present a family of perfect quantum strategies in any $(2,d-1,d)$ Bell scenario, with $d = 2^kp^m$ for $p$ prime, $m \geq k \geq 0$ (excluding $m=k=0$), $d = 8p$ for $p \geq 19$, $d=kp$ for $p > ((k-2)2^{k-2})^2$ whenever there exists a Hadamard matrix of order $k$, other sporadic examples, as well as a recursive construction that produces perfect quantum strategies for infinitely many dimensions $d$ from any dimension $d'$ with a perfect quantum strategy. We identify their associated Bell inequalities and prove that they are not tight, which provides a second counterexample to a conjecture of 2007., Comment: 26 pages, 5 figures

Let $F$ be the finite field of order $q$ and $\M(n,r, F)$ be the set of $n\times n$ matrices of rank $r$ over the field $F$. For $\alpha\in F$ and $A\in \M(n,F)$, let $$Z^{\alpha}_{A,r}=\left\{X\in \M(n,r, F)\mid \tr(AX)=\alpha\right \}.$$ In this article, we solve the problem of determining the cardinality of $Z_{A,r}^{\alpha}$. We also solve the generalization of the problem to rectangular matrices.

Quantum nonlocality based on state discrimination describes the global property of the set of orthogonal states and has a wide range of applications in quantum cryptographic protocols. Strongest nonlocality is the strongest form of quantum nonlocality recently presented in multipartite quantum systems: a set of orthogonal multipartite quantum states is strongest nonlocal if the only orthogonality-preserving local measurements on the subsystems in every bipartition are trivial. In this work, we found a construction of strongest nonlocal sets in $\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}$ $(2\leq d_{1}\leq d_{2}\leq d_{3})$ of size $d_2d_3+1$ without stopper states. Then we obtain the strongest nonlocal sets in four-partite systems with $d^3+1$ orthogonal states in $\mathbb{C}^d\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$ $(d\geq2)$ and $d_{2}d_{3}d_{4}+1$ orthogonal states in $\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}\otimes \mathbb{C}^{d_{4}}$ $(2\leq d_{1}\leq d_{2}\leq d_{3}\leq d_{4})$. Surprisingly, the number of the elements in all above constructions perfectly reaches the recent conjectured lower bound and reduces the size of the strongest nonlocal set in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$ of [\href{https://doi.org/10.1103/PhysRevA.108.062407}{Phys. Rev. A \textbf{108}, 062407 (2023)}] by $d-2$. In particular, the general optimal construction of the strongest nonlocal set in four-partite system is completely solved for the first time, which further highlights the theory of quantum nonlocality from the perspective of state discrimination., Comment: 14 pages, 3 figures

Object-centric scene decompositions are important representations for downstream tasks in fields such as computer vision and robotics. The recently proposed Slot Attention module, already leveraged by several derivative works for image segmentation and object tracking in videos, is a deep learning component which performs unsupervised object-centric scene decomposition on input images. It is based on an attention architecture, in which latent slot vectors, which hold compressed information on objects, attend to localized perceptual features from the input image. In this paper, we demonstrate that design decisions on normalizing the aggregated values in the attention architecture have considerable impact on the capabilities of Slot Attention to generalize to a higher number of slots and objects as seen during training. We propose and investigate alternatives to the original normalization scheme which increase the generalization capabilities of Slot Attention to varying slot and object counts, resulting in performance gains on the task of unsupervised image segmentation. The newly proposed normalizations represent minimal and easy to implement modifications of the usual Slot Attention module, changing the value aggregation mechanism from a weighted mean operation to a scaled weighted sum operation., Comment: 30 pages

Cardinality sketches are popular data structures that enhance the efficiency of working with large data sets. The sketches are randomized representations of sets that are only of logarithmic size but can support set merges and approximate cardinality (i.e., distinct count) queries. When queries are not adaptive, that is, they do not depend on preceding query responses, the design provides strong guarantees of correctly answering a number of queries exponential in the sketch size $k$. In this work, we investigate the performance of cardinality sketches in adaptive settings and unveil inherent vulnerabilities. We design an attack against the ``standard'' estimators that constructs an adversarial input by post-processing responses to a set of simple non-adaptive queries of size linear in the sketch size $k$. Empirically, our attack used only $4k$ queries with the widely used HyperLogLog (HLL++)~\citep{hyperloglog:2007,hyperloglogpractice:EDBT2013} sketch. The simple attack technique suggests it can be effective with post-processed natural workloads. Finally and importantly, we demonstrate that the vulnerability is inherent as \emph{any} estimator applied to known sketch structures can be attacked using a number of queries that is quadratic in $k$, matching a generic upper bound.

Cardinality Estimation (CE) for query is to estimate the number of results without execution, which is an effective index in query optimization. Recently, CE for queries over knowlege graph (KGs) with triple facts has achieved great success. To more precisely represent facts, current researchers propose hyper-relational KGs (HKGs) to represent a triple fact with qualifiers providing additional context to the fact. However, existing CE methods, such as sampling and summary methods over KGs, perform unsatisfactorily on HKGs due to the complexity of qualifiers. Learning-based CE methods do not utilize qualifier information to learn query representation accurately, leading to poor performance. Also, there is only one limited CE benchmark for HKG query, which is not comprehensive and only covers limited patterns. The lack of querysets over HKG also becomes a bottleneck to comprehensively investigate CE problems on HKGs. In this work, we first construct diverse and unbiased hyper-relational querysets over three popular HKGs for investigating CE. Besides, we also propose a novel qualifier-aware graph neural network (GNN) model that effectively incorporates qualifier information and adaptively combines outputs from multiple GNN layers, to accurately predict the cardinality. Our experiments demonstrate that our model outperforms all state-of-the-art CE methods over three benchmarks on popular HKGs.

Graph workloads pose a particularly challenging problem for query optimizers. They typically feature large queries made up of entirely many-to-many joins with complex correlations. This puts significant stress on traditional cardinality estimation methods which generally see catastrophic errors when estimating the size of queries with only a handful of joins. To overcome this, we propose COLOR, a framework for subgraph cardinality estimation which applies insights from graph compression theory to produce a compact summary that captures the global topology of the data graph. Further, we identify several key optimizations that enable tractable estimation over this summary even for large query graphs. We then evaluate several designs within this framework and find that they improve accuracy by up to 10$^3$x over all competing methods while maintaining fast inference, a small memory footprint, efficient construction, and graceful degradation under updates.

In this paper, we study the embedded feature selection problem in linear Support Vector Machines (SVMs), in which a cardinality constraint is employed, leading to an interpretable classification model. The problem is NP-hard due to the presence of the cardinality constraint, even though the original linear SVM amounts to a problem solvable in polynomial time. To handle the hard problem, we first introduce two mixed-integer formulations for which novel semidefinite relaxations are proposed. Exploiting the sparsity pattern of the relaxations, we decompose the problems and obtain equivalent relaxations in a much smaller cone, making the conic approaches scalable. To make the best usage of the decomposed relaxations, we propose heuristics using the information of its optimal solution. Moreover, an exact procedure is proposed by solving a sequence of mixed-integer decomposed semidefinite optimization problems. Numerical results on classical benchmarking datasets are reported, showing the efficiency and effectiveness of our approach., Comment: Submitted to European Journal of Operational Research. arXiv admin note: text overlap with arXiv:1808.02435 by other authors