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We develop a framework for incorporating edge-dependent vertex weights (EDVWs) into the hypergraph minimum s-t cut problem. These weights are able to reflect different importance of vertices within a hyperedge, thus leading to better characterized cut properties. More precisely, we introduce a new class of hyperedge splitting functions that we call EDVWs-based, where the penalty of splitting a hyperedge depends only on the sum of EDVWs associated with the vertices on each side of the split. Moreover, we provide a way to construct submodular EDVWs-based splitting functions and prove that a hypergraph equipped with such splitting functions can be reduced to a graph sharing the same cut properties. In this case, the hypergraph minimum s-t cut problem can be solved using well-developed solutions to the graph minimum s-t cut problem. In addition, we show that an existing sparsification technique can be easily extended to our case and makes the reduced graph smaller and sparser, thus further accelerating the algorithms applied to the reduced graph. Numerical experiments using real-world data demonstrate the effectiveness of our proposed EDVWs-based splitting functions in comparison with the all-or-nothing splitting function and cardinality-based splitting functions commonly adopted in existing work.

We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed $\alpha$ between zero and one we are given a graph and two numbers $k \in \mathbb{N}$ and $t \in \mathbb{Q}$. The task is to find a vertex subset $S$ of exactly $k$ vertices that has value at least (resp. at most for minimization) $t$. Here, the value of a vertex set computes as $\alpha$ times the number of edges with exactly one endpoint in $S$ plus $1-\alpha$ times the number of edges with both endpoints in $S$. These two problems generalize many prominent graph problems, such as Densest $k$-Subgraph, Sparsest $k$-Subgraph, Partial Vertex Cover, and Max ($k$,$n-k$)-Cut. In this work, we complete the picture of their parameterized complexity on several types of sparse graphs that are described by structural parameters. In particular, we provide kernelization algorithms and kernel lower bounds for these problems. A somewhat surprising consequence of our kernelizations is that Partial Vertex Cover and Max $(k,n-k)$-Cut not only behave in the same way but that the kernels for both problems can be obtained by the same algorithms.

We consider the problem of computing an $(s,d)$-hypernetwork in an acyclic F-hypergraph. This is a fundamental computational problem arising in directed hypergraphs, and is a foundational step in tackling problems of reachability and redundancy. This problem was previously explored in the context of general directed hypergraphs (containing cycles), where it is NP-hard, and acyclic B-hypergraphs, where a linear time algorithm can be achieved. In a surprising contrast, we find that for acyclic F-hypergraphs the problem is NP-hard, which also implies the problem is hard in BF-hypergraphs. This is a striking complexity boundary given that F-hypergraphs and B-hypergraphs would at first seem to be symmetrical to one another. We provide the proof of complexity and explain why there is a fundamental asymmetry between the two classes of directed hypergraphs.

In this paper, we study the \emph{type graph}, namely a bipartite graph induced by a joint type. We investigate the maximum edge density of induced bipartite subgraphs of this graph having a number of vertices on each side on an exponential scale in the length $n$ of the type. This can be seen as an isoperimetric problem. We provide asymptotically sharp bounds for the exponent of the maximum edge density as the length of the type goes to infinity. We also study the biclique rate region of the type graph, which is defined as the set of $\left(R_{1},R_{2}\right)$ such that there exists a biclique of the type graph which has respectively $e^{nR_{1}}$ and $e^{nR_{2}}$ vertices on the two sides. We provide asymptotically sharp bounds for the biclique rate region as well. We then apply our results and proof ideas to noninteractive simulation problems. We completely characterize the exponents of maximum and minimum joint probabilities when the marginal probabilities vanish exponentially fast with given exponents. These results can be seen as strong small-set expansion theorems. We extend the noninteractive simulation problem by replacing Boolean functions with arbitrary nonnegative functions, and obtain new hypercontractivity inequalities which are stronger than the common hypercontractivity inequalities. Furthermore, as an application of our results, a new outer bound for the zero-error capacity region of the binary adder channel is provided, which improves the previously best known bound, due to Austrin, Kaski, Koivisto, and Nederlof. Our proofs in this paper are based on the method of types, linear algebra, and coupling techniques.

Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.

Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.

We study the problem of embedding-based entity alignment between knowledge graphs (KGs). Previous works mainly focus on the relational structure of entities. Some further incorporate another type of features, such as attributes, for refinement. However, a vast of entity features are still unexplored or not equally treated together, which impairs the accuracy and robustness of embedding-based entity alignment. In this paper, we propose a novel framework that unifies multiple views of entities to learn embeddings for entity alignment. Specifically, we embed entities based on the views of entity names, relations and attributes, with several combination strategies. Furthermore, we design some cross-KG inference methods to enhance the alignment between two KGs. Our experiments on real-world datasets show that the proposed framework significantly outperforms the state-of-the-art embedding-based entity alignment methods. The selected views, cross-KG inference and combination strategies all contribute to the performance improvement.

Knowledge graphs are large graph-structured databases of facts, which typically suffer from incompleteness. Link prediction is the task of inferring missing relations (links) between entities (nodes) in a knowledge graph. We approach this task using a hypernetwork architecture to generate convolutional layer filters specific to each relation and apply those filters to the subject entity embeddings. This architecture enables a trade-off between non-linear expressiveness and the number of parameters to learn. Our model simplifies the entity and relation embedding interactions introduced by the predecessor convolutional model, while outperforming all previous approaches to link prediction across all standard link prediction datasets.

Meta-graph is currently the most powerful tool for similarity search on heterogeneous information networks,where a meta-graph is a composition of meta-paths that captures the complex structural information. However, current relevance computing based on meta-graph only considers the complex structural information, but ignores its embedded meta-paths information. To address this problem, we proposeMEta-GrAph-based network embedding models, called MEGA and MEGA++, respectively. The MEGA model uses normalized relevance or similarity measures that are derived from a meta-graph and its embedded meta-paths between nodes simultaneously, and then leverages tensor decomposition method to perform node embedding. The MEGA++ further facilitates the use of coupled tensor-matrix decomposition method to obtain a joint embedding for nodes, which simultaneously considers the hidden relations of all meta information of a meta-graph.Extensive experiments on two real datasets demonstrate thatMEGA and MEGA++ are more effective than state-of-the-art approaches.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.