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The hard thresholding technique plays a vital role in the development of algorithms for sparse signal recovery. By merging this technique and heavy-ball acceleration method which is a multi-step extension of the traditional gradient descent method, we propose the so-called heavy-ball-based hard thresholding (HBHT) and heavy-ball-based hard thresholding pursuit (HBHTP) algorithms for signal recovery. It turns out that the HBHT and HBHTP can successfully recover a $k$-sparse signal if the restricted isometry constant of the measurement matrix satisfies $\delta_{3k}<0.618 $ and $\delta_{3k}<0.577,$ respectively. The guaranteed success of HBHT and HBHTP is also shown under the conditions $\delta_{2k}<0.356$ and $\delta_{2k}<0.377,$ respectively. Moreover, the finite convergence and stability of the two algorithms are also established in this paper. Simulations on random problem instances are performed to compare the performance of the proposed algorithms and several existing ones. Empirical results indicate that the HBHTP performs very comparably to a few existing algorithms and it takes less average time to achieve the signal recovery than these existing methods.

While algorithms for planar graphs have received a lot of attention, few papers have focused on the additional power that one gets from assuming an embedding of the graph is available. While in the classic sequential setting, this assumption gives no additional power (as a planar graph can be embedded in linear time), we show that this is far from being the case in other settings. We assume that the embedding is straight-line, but our methods also generalize to non-straight-line embeddings. Specifically, we focus on sublinear-time computation and massively parallel computation (MPC). Our main technical contribution is a sublinear-time algorithm for computing a relaxed version of an $r$-division. We then show how this can be used to estimate Lipschitz additive graph parameters. This includes, for example, the maximum matching, maximum independent set, or the minimum dominating set. We also show how this can be used to solve some property testing problems with respect to the vertex edit distance. In the second part of our paper, we show an MPC algorithm that computes an $r$-division of the input graph. We show how this can be used to solve various classical graph problems with space per machine of $O(n^{2/3+\epsilon})$ for some $\epsilon>0$, and while performing $O(1)$ rounds. This includes for example approximate shortest paths or the minimum spanning tree. Our results also imply an improved MPC algorithm for Euclidean minimum spanning tree.

A directed graph is oriented if it can be obtained by orienting the edges of a simple, undirected graph. For an oriented graph $G$, let $\beta(G)$ denote the size of a minimum feedback arc set, a smallest subset of edges whose deletion leaves an acyclic subgraph. A simple consequence of a result of Berger and Shor is that any oriented graph $G$ with $m$ edges satisfies $\beta(G) = m/2 - \Omega(m^{3/4})$. We observe that if an oriented graph $G$ has a fixed forbidden subgraph $B$, the upper bound of $\beta(G) = m/2 - \Omega(m^{3/4})$ is best possible as a function of the number of edges if $B$ is not bipartite, but the exponent $3/4$ in the lower order term can be improved if $B$ is bipartite. We also show that for every rational number $r$ between $3/4$ and $1$, there is a finite collection of digraphs $\mathcal{B}$ such that every $\mathcal{B}$-free digraph $G$ with $m$ edges satisfies $\beta(G) = m/2 - \Omega(m^r)$, and this bound is best possible up to the implied constant factor. The proof uses a connection to Tur\'an numbers and a result of Bukh and Conlon. Both of our upper bounds come equipped with randomized linear-time algorithms that construct feedback arc sets achieving those bounds. Finally, we give a characterization of quasirandom directed graphs via minimum feedback arc sets.

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to $H$-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a {\em dynamic setting}, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of $(1+\epsilon)r^2$ and an update time of $O(\text{poly} (r, \log n))$, where $r$ denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of $(1+\epsilon)$ that is independent of $r$, and a similar update time of $O(\text{poly} (r, \log n))$. It is the first $(1+\epsilon)$-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.

A palindromic substring $T[i.. j]$ of a string $T$ is said to be a shortest unique palindromic substring (SUPS) in $T$ for an interval $[p, q]$ if $T[i.. j]$ is a shortest one such that $T[i.. j]$ occurs only once in $T$, and $[i, j]$ contains $[p, q]$. The SUPS problem is, given a string $T$ of length $n$, to construct a data structure that can compute all the SUPSs for any given query interval. It is known that any SUPS query can be answered in $O(\alpha)$ time after $O(n)$-time preprocessing, where $\alpha$ is the number of SUPSs to output [Inoue et al., 2018]. In this paper, we first show that $\alpha$ is at most $4$, and the upper bound is tight. Also, we present an algorithm to solve the SUPS problem for a sliding window that can answer any query in $O(\log\log W)$ time and update data structures in amortized $O(\log\sigma)$ time, where $W$ is the size of the window, and $\sigma$ is the alphabet size. Furthermore, we consider the SUPS problem in the after-edit model and present an efficient algorithm. Namely, we present an algorithm that uses $O(n)$ time for preprocessing and answers any $k$ SUPS queries in $O(\log n\log\log n + k\log\log n)$ time after single character substitution. As a by-product, we propose a fully-dynamic data structure for range minimum queries (RmQs) with a constraint where the width of each query range is limited to polylogarithmic. The constrained RmQ data structure can answer such a query in constant time and support a single-element edit operation in amortized constant time.

Spectral clustering (SC) is a popular clustering technique to find strongly connected communities on a graph. SC can be used in Graph Neural Networks (GNNs) to implement pooling operations that aggregate nodes belonging to the same cluster. However, the eigendecomposition of the Laplacian is expensive and, since clustering results are graph-specific, pooling methods based on SC must perform a new optimization for each new sample. In this paper, we propose a graph clustering approach that addresses these limitations of SC. We formulate a continuous relaxation of the normalized minCUT problem and train a GNN to compute cluster assignments that minimize this objective. Our GNN-based implementation is differentiable, does not require to compute the spectral decomposition, and learns a clustering function that can be quickly evaluated on out-of-sample graphs. From the proposed clustering method, we design a graph pooling operator that overcomes some important limitations of state-of-the-art graph pooling techniques and achieves the best performance in several supervised and unsupervised tasks.

Translational distance-based knowledge graph embedding has shown progressive improvements on the link prediction task, from TransE to the latest state-of-the-art RotatE. However, N-1, 1-N and N-N predictions still remain challenging. In this work, we propose a novel translational distance-based approach for knowledge graph link prediction. The proposed method includes two-folds, first we extend the RotatE from 2D complex domain to high dimension space with orthogonal transforms to model relations for better modeling capacity. Second, the graph context is explicitly modeled via two directed context representations. These context representations are used as part of the distance scoring function to measure the plausibility of the triples during training and inference. The proposed approach effectively improves prediction accuracy on the difficult N-1, 1-N and N-N cases for knowledge graph link prediction task. The experimental results show that it achieves better performance on two benchmark data sets compared to the baseline RotatE, especially on data set (FB15k-237) with many high in-degree connection nodes.

In this paper we provide a comprehensive introduction to knowledge graphs, which have recently garnered significant attention from both industry and academia in scenarios that require exploiting diverse, dynamic, large-scale collections of data. After a general introduction, we motivate and contrast various graph-based data models and query languages that are used for knowledge graphs. We discuss the roles of schema, identity, and context in knowledge graphs. We explain how knowledge can be represented and extracted using a combination of deductive and inductive techniques. We summarise methods for the creation, enrichment, quality assessment, refinement, and publication of knowledge graphs. We provide an overview of prominent open knowledge graphs and enterprise knowledge graphs, their applications, and how they use the aforementioned techniques. We conclude with high-level future research directions for knowledge graphs.

The recent proliferation of knowledge graphs (KGs) coupled with incomplete or partial information, in the form of missing relations (links) between entities, has fueled a lot of research on knowledge base completion (also known as relation prediction). Several recent works suggest that convolutional neural network (CNN) based models generate richer and more expressive feature embeddings and hence also perform well on relation prediction. However, we observe that these KG embeddings treat triples independently and thus fail to cover the complex and hidden information that is inherently implicit in the local neighborhood surrounding a triple. To this effect, our paper proposes a novel attention based feature embedding that captures both entity and relation features in any given entity's neighborhood. Additionally, we also encapsulate relation clusters and multihop relations in our model. Our empirical study offers insights into the efficacy of our attention based model and we show marked performance gains in comparison to state of the art methods on all datasets.