Maximum bipartite matching (MBM) is a fundamental problem in combinatorial optimization with a long and rich history. A classic result of Hopcroft and Karp (1973) provides an $O(m \sqrt{n})$-time algorithm for the problem, where $n$ and $m$ are the number of vertices and edges in the input graph, respectively. For dense graphs, an approach based on fast matrix multiplication achieves a running time of $O(n^{2.371})$. For several decades, these results represented state-of-the-art algorithms, until, in 2013, Madry introduced a powerful new approach for solving MBM using continuous optimization techniques. This line of research led to several spectacular results, culminating in a breakthrough $m^{1+o(1)}$-time algorithm for min-cost flow, that implies an $m^{1+o(1)}$-time algorithm for MBM as well. These striking advances naturally raise the question of whether combinatorial algorithms can match the performance of the algorithms that are based on continuous techniques for MBM. A recent work of the authors (2024) made progress on this question by giving a combinatorial $\tilde{O}(m^{1/3}n^{5/3})$-time algorithm for MBM, thus outperforming both the Hopcroft-Karp algorithm and matrix multiplication based approaches, on sufficiently dense graphs. Still, a large gap remains between the running time of their algorithm and the almost linear-time achievable by algorithms based on continuous techniques. In this work, we take another step towards narrowing this gap, and present a randomized $n^{2+o(1)}$-time combinatorial algorithm for MBM. Thus in dense graphs, our algorithm essentially matches the performance of algorithms that are based on continuous methods. We also obtain a randomized $n^{2+o(1)}$-time combinatorial algorithm for maximum vertex-capacitated $s$-$t$ flow in directed graphs when all vertex capacities are identical, using a standard reduction from this problem to MBM.
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讓 iOS 8 和 OS X Yosemite 無縫切換的一個新特性。
> Apple products have always been designed to work together beautifully. But now they may really surprise you. With iOS 8 and OS X Yosemite, you’ll be able to do more wonderful things than ever before.
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Graph neural networks form a class of deep learning architectures specifically designed to work with graph-structured data. As such, they share the inherent limitations and problems of deep learning, especially regarding the issues of explainability and trustworthiness. We propose $\mu\mathcal{G}$, an original domain-specific language for the specification of graph neural networks that aims to overcome these issues. The language's syntax is introduced, and its meaning is rigorously defined by a denotational semantics. An equivalent characterization in the form of an operational semantics is also provided and, together with a type system, is used to prove the type soundness of $\mu\mathcal{G}$. We show how $\mu\mathcal{G}$ programs can be represented in a more user-friendly graphical visualization, and provide examples of its generality by showing how it can be used to define some of the most popular graph neural network models, or to develop any custom graph processing application.
Parameterized Inapproximability Hypothesis (PIH) is a central question in the field of parameterized complexity. PIH asserts that given as input a 2-CSP on $k$ variables and alphabet size $n$, it is W[1]-hard parameterized by $k$ to distinguish if the input is perfectly satisfiable or if every assignment to the input violates 1% of the constraints. An important implication of PIH is that it yields the tight parameterized inapproximability of the $k$-maxcoverage problem. In the $k$-maxcoverage problem, we are given as input a set system, a threshold $\tau>0$, and a parameter $k$ and the goal is to determine if there exist $k$ sets in the input whose union is at least $\tau$ fraction of the entire universe. PIH is known to imply that it is W[1]-hard parameterized by $k$ to distinguish if there are $k$ input sets whose union is at least $\tau$ fraction of the universe or if the union of every $k$ input sets is not much larger than $\tau\cdot (1-\frac{1}{e})$ fraction of the universe. In this work we present a gap preserving FPT reduction (in the reverse direction) from the $k$-maxcoverage problem to the aforementioned 2-CSP problem, thus showing that the assertion that approximating the $k$-maxcoverage problem to some constant factor is W[1]-hard implies PIH. In addition, we present a gap preserving FPT reduction from the $k$-median problem (in general metrics) to the $k$-maxcoverage problem, further highlighting the power of gap preserving FPT reductions over classical gap preserving polynomial time reductions.
In the metric distortion problem there is a set of candidates $C$ and voters $V$ in the same metric space. The goal is to select a candidate minimizing the social cost: the sum of distances of the selected candidate from all the voters, and the challenge arises from the algorithm receiving only ordinaL input: each voter's ranking of candidate, while the objective function is cardinal, determined by the underlying metric. The distortion of an algorithm is its worst-case approximation factor of the optimal social cost. A key concept here is the (p,q)-veto core, with $p\in \Delta(V)$ and $q\in \Delta(C)$ being normalized weight vectors representing voters' veto power and candidates' support, respectively. The (p,q)-veto core corresponds to a set of winners from a specific class of deterministic algorithms. Notably, the optimal distortion of $3$ is obtained from this class, by selecting veto core candidates using uniform $p$ and $q$ proportional to candidates' plurality scores. Bounding the distortion of other algorithms from this class is an open problem. Our contribution is twofold. First, we establish upper bounds on the distortion of candidates from the (p,q)-veto core for arbitrary weight vectors $p$ and $q$. Second, we revisit the metric distortion problem through the \emph{learning-augmented} framework, which equips the algorithm with a (machine-learned) prediction regarding the optimal candidate. The quality of this prediction is unknown, and the goal is to optimize the algorithm's performance under accurate predictions (consistency), while simultaneously providing worst-case guarantees under arbitrarily inaccurate predictions (robustness). We propose an algorithm that chooses candidates from the (p,q)-veto core, using a prediction-guided q vector and, leveraging our distortion bounds, we prove that this algorithm achieves the optimal robustness-consistency trade-off.
Error-bounded lossy compression is a critical technique for significantly reducing scientific data volumes. Compared to CPU-based compressors, GPU-based compressors exhibit substantially higher throughputs, fitting better for today's HPC applications. However, the critical limitations of existing GPU-based compressors are their low compression ratios and qualities, severely restricting their applicability. To overcome these, we introduce a new GPU-based error-bounded scientific lossy compressor named cuSZ-$i$, with the following contributions: (1) A novel GPU-optimized interpolation-based prediction method significantly improves the compression ratio and decompression data quality. (2) The Huffman encoding module in cuSZ-$i$ is optimized for better efficiency. (3) cuSZ-$i$ is the first to integrate the NVIDIA Bitcomp-lossless as an additional compression-ratio-enhancing module. Evaluations show that cuSZ-$i$ significantly outperforms other latest GPU-based lossy compressors in compression ratio under the same error bound (hence, the desired quality), showcasing a 476% advantage over the second-best. This leads to cuSZ-$i$'s optimized performance in several real-world use cases.
We develop a novel clustering method for distributional data, where each data point is regarded as a probability distribution on the real line. For distributional data, it has been challenging to develop a clustering method that utilizes the mode of variation of data because the space of probability distributions lacks a vector space structure, preventing the application of existing methods for functional data. In this study, we propose a novel clustering method for distributional data on the real line, which takes account of difference in both the mean and mode of variation structures of clusters, in the spirit of the $k$-centres clustering approach proposed for functional data. Specifically, we consider the space of distributions equipped with the Wasserstein metric and define a geodesic mode of variation of distributional data using geodesic principal component analysis. Then, we utilize the geodesic mode of each cluster to predict the cluster membership of each distribution. We theoretically show the validity of the proposed clustering criterion by studying the probability of correct membership. Through a simulation study and real data application, we demonstrate that the proposed distributional clustering method can improve cluster quality compared to conventional clustering algorithms.
When verifying liveness properties on a transition system, it is often necessary to discard spurious violating paths by making assumptions on which paths represent realistic executions. Capturing that some property holds under such an assumption in a logical formula is challenging and error-prone, particularly in the modal $\mu$-calculus. In this paper, we present template formulae in the modal $\mu$-calculus that can be instantiated to a broad range of liveness properties. We consider the following assumptions: progress, justness, weak fairness, strong fairness, and hyperfairness, each with respect to actions. The correctness of these formulae has been proven.
We develop domain theory in constructive and predicative univalent foundations (also known as homotopy type theory). That we work predicatively means that we do not assume Voevodsky's propositional resizing axioms. Our work is constructive in the sense that we do not rely on excluded middle or the axiom of (countable) choice. Domain theory studies so-called directed complete posets (dcpos) and Scott continuous maps between them and has applications in a variety of fields, such as programming language semantics, higher-type computability and topology. A common approach to deal with size issues in a predicative foundation is to work with information systems, abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. In our type-theoretic approach, we instead accept that dcpos may be large and work with type universes to account for this. A priori one might expect that iterative constructions of dcpos may result in a need for ever-increasing universes and are predicatively impossible. We show, through a careful tracking of type universe parameters, that such constructions can be carried out in a predicative setting. In particular, we give a predicative reconstruction of Scott's $D_\infty$ model of the untyped $\lambda$-calculus. Our work is formalised in the Agda proof assistant and its ability to infer universe levels has been invaluable for our purposes.
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.
We present a novel data-driven strategy to choose the hyperparameter $k$ in the $k$-NN regression estimator without using any hold-out data. We treat the problem of choosing the hyperparameter as an iterative procedure (over $k$) and propose using an easily implemented in practice strategy based on the idea of early stopping and the minimum discrepancy principle. This model selection strategy is proven to be minimax-optimal over some smoothness function classes, for instance, the Lipschitz functions class on a bounded domain. The novel method often improves statistical performance on artificial and real-world data sets in comparison to other model selection strategies, such as the Hold-out method, 5-fold cross-validation, and AIC criterion. The novelty of the strategy comes from reducing the computational time of the model selection procedure while preserving the statistical (minimax) optimality of the resulting estimator. More precisely, given a sample of size $n$, if one should choose $k$ among $\left\{ 1, \ldots, n \right\}$, and $\left\{ f^1, \ldots, f^n \right\}$ are the estimators of the regression function, the minimum discrepancy principle requires the calculation of a fraction of the estimators, while this is not the case for the generalized cross-validation, Akaike's AIC criteria, or Lepskii principle.
Click-through rate (CTR) prediction plays a critical role in recommender systems and online advertising. The data used in these applications are multi-field categorical data, where each feature belongs to one field. Field information is proved to be important and there are several works considering fields in their models. In this paper, we proposed a novel approach to model the field information effectively and efficiently. The proposed approach is a direct improvement of FwFM, and is named as Field-matrixed Factorization Machines (FmFM, or $FM^2$). We also proposed a new explanation of FM and FwFM within the FmFM framework, and compared it with the FFM. Besides pruning the cross terms, our model supports field-specific variable dimensions of embedding vectors, which acts as soft pruning. We also proposed an efficient way to minimize the dimension while keeping the model performance. The FmFM model can also be optimized further by caching the intermediate vectors, and it only takes thousands of floating-point operations (FLOPs) to make a prediction. Our experiment results show that it can out-perform the FFM, which is more complex. The FmFM model's performance is also comparable to DNN models which require much more FLOPs in runtime.
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