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Majority-SAT is the problem of determining whether an input $n$-variable formula in conjunctive normal form (CNF) has at least $2^{n-1}$ satisfying assignments. Majority-SAT and related problems have been studied extensively in various AI communities interested in the complexity of probabilistic planning and inference. Although Majority-SAT has been known to be PP-complete for over 40 years, the complexity of a natural variant has remained open: Majority-$k$SAT, where the input CNF formula is restricted to have clause width at most $k$. We prove that for every $k$, Majority-$k$SAT is in P. In fact, for any positive integer $k$ and rational $\rho \in (0,1)$ with bounded denominator, we give an algorithm that can determine whether a given $k$-CNF has at least $\rho \cdot 2^n$ satisfying assignments, in deterministic linear time (whereas the previous best-known algorithm ran in exponential time). Our algorithms have interesting positive implications for counting complexity and the complexity of inference, significantly reducing the known complexities of related problems such as E-MAJ-$k$SAT and MAJ-MAJ-$k$SAT. At the heart of our approach is an efficient method for solving threshold counting problems by extracting sunflowers found in the corresponding set system of a $k$-CNF. We also show that the tractability of Majority-$k$SAT is somewhat fragile. For the closely related GtMajority-SAT problem (where we ask whether a given formula has greater than $2^{n-1}$ satisfying assignments) which is known to be PP-complete, we show that GtMajority-$k$SAT is in P for $k\le 3$, but becomes NP-complete for $k\geq 4$. These results are counterintuitive, because the ``natural'' classifications of these problems would have been PP-completeness, and because there is a stark difference in the complexity of GtMajority-$k$SAT and Majority-$k$SAT for all $k\ge 4$.

With the rapid development of deep learning techniques, various recent work has tried to apply graph neural networks (GNNs) to solve NP-hard problems such as Boolean Satisfiability (SAT), which shows the potential in bridging the gap between machine learning and symbolic reasoning. However, the quality of solutions predicted by GNNs has not been well investigated in the literature. In this paper, we study the capability of GNNs in learning to solve Maximum Satisfiability (MaxSAT) problem, both from theoretical and practical perspectives. We build two kinds of GNN models to learn the solution of MaxSAT instances from benchmarks, and show that GNNs have attractive potential to solve MaxSAT problem through experimental evaluation. We also present a theoretical explanation of the effect that GNNs can learn to solve MaxSAT problem to some extent for the first time, based on the algorithmic alignment theory.

The hop-constrained Steiner tree problem (HSTP) is a generalization of the classical Steiner tree problem. It asks for a minimum cost subtree that spans some specified nodes of a given graph, such that the number of edges between each node of the tree and its root respects a given hop limit. This NP-hard problem has many variants, often modeled as integer linear programs. Two of the models are so-called assignment and partial-ordering based models, which yield (up to our knowledge) the best two state-of-the-art formulations for the variant Steiner tree problem with revenues, budgets, and hop constraints (STPRBH). The solution of the HSTP and its variants such as the STPRBH and the hop-constrained minimum spanning tree problem (HMSTP) is a hop-constrained tree, a rooted tree whose depth is bounded by a given hop limit. This paper provides some theoretical results that show the polyhedral advantages of the partial-ordering model over the assignment model for this class of problems. Computational results in this paper and the literature for the HSTP, STPRBH, and HMSTP show that the partial-ordering model outperforms the assignment model in practice, too; it has better linear programming relaxation and solves more instances.

In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an elegant form in the case of Toeplitz matrices generated by a scalar-valued function. In the block-Toeplitz setting, that is, in the case where the matrix entries are small generic matrices instead of scalars, some algorithms have already been proposed regarding specific applications and a first rigorous convergence analysis has been performed in [7]. However, with the existent symbol-based theoretical tools, it is still not possible to prove the convergence of many multigrid methods known in the literature. This paper aims to generalize the previous results giving more general sufficient conditions on the symbol of the grid transfer operators.In particular, we treat matrix-valued trigonometric polynomials which can be non-diagonalizable and singular at all points and we express the new conditions in terms of the eigenvectors associated with the ill-conditioned subspace. Moreover, we extend the analysis to the V-cycle method proving a linear convergence rate under stronger conditions, which resemble those given in the scalar case. In order to validate our theoretical findings, we present a classical block structured problem stemming from a FEM approximation of a second order differential problem. We focus on two multigrid strategies that use the geometric and the standard bisection grid transfer operators and we prove that both fall into the category of projectors satisfying the proposed conditions. In addition, using a tensor product argument, we provide a strategy to construct efficient V-cycle procedures in the block multilevel setting.

We present new greedy and beam search heuristic methods to find small-size $k$-dominating sets in graphs. The methods are inspired by a new problem formulation which explicitly highlights a certain structure of the problem. An empirical evaluation of the new methods is done with respect to two existing methods, using instances of graphs corresponding to street networks. The k-domination problem with respect to this class of graphs can be used to model real-world facility location problem scenarios. For the classic minimum dominating set ($1$-domination) problem, all except one methods perform similarly, which is due to their equivalence in this particular case. However, for the k-domination problem with k>1, the new methods outperform the benchmark methods, and the performance gain is more significant for larger values of k.

Graph Convolutional Networks (GCNs) have been widely used due to their outstanding performance in processing graph-structured data. However, the undirected graphs limit their application scope. In this paper, we extend spectral-based graph convolution to directed graphs by using first- and second-order proximity, which can not only retain the connection properties of the directed graph, but also expand the receptive field of the convolution operation. A new GCN model, called DGCN, is then designed to learn representations on the directed graph, leveraging both the first- and second-order proximity information. We empirically show the fact that GCNs working only with DGCNs can encode more useful information from graph and help achieve better performance when generalized to other models. Moreover, extensive experiments on citation networks and co-purchase datasets demonstrate the superiority of our model against the state-of-the-art methods.

Graph representation learning resurges as a trending research subject owing to the widespread use of deep learning for Euclidean data, which inspire various creative designs of neural networks in the non-Euclidean domain, particularly graphs. With the success of these graph neural networks (GNN) in the static setting, we approach further practical scenarios where the graph dynamically evolves. Existing approaches typically resort to node embeddings and use a recurrent neural network (RNN, broadly speaking) to regulate the embeddings and learn the temporal dynamics. These methods require the knowledge of a node in the full time span (including both training and testing) and are less applicable to the frequent change of the node set. In some extreme scenarios, the node sets at different time steps may completely differ. To resolve this challenge, we propose EvolveGCN, which adapts the graph convolutional network (GCN) model along the temporal dimension without resorting to node embeddings. The proposed approach captures the dynamism of the graph sequence through using an RNN to evolve the GCN parameters. Two architectures are considered for the parameter evolution. We evaluate the proposed approach on tasks including link prediction, edge classification, and node classification. The experimental results indicate a generally higher performance of EvolveGCN compared with related approaches. The code is available at \url{//github.com/IBM/EvolveGCN}.

Modeling generative process of growing graphs has wide applications in social networks and recommendation systems, where cold start problem leads to new nodes isolated from existing graph. Despite the emerging literature in learning graph representation and graph generation, most of them can not handle isolated new nodes without nontrivial modifications. The challenge arises due to the fact that learning to generate representations for nodes in observed graph relies heavily on topological features, whereas for new nodes only node attributes are available. Here we propose a unified generative graph convolutional network that learns node representations for all nodes adaptively in a generative model framework, by sampling graph generation sequences constructed from observed graph data. We optimize over a variational lower bound that consists of a graph reconstruction term and an adaptive Kullback-Leibler divergence regularization term. We demonstrate the superior performance of our approach on several benchmark citation network datasets.

Recent advancements in deep neural networks for graph-structured data have led to state-of-the-art performance on recommender system benchmarks. However, making these methods practical and scalable to web-scale recommendation tasks with billions of items and hundreds of millions of users remains a challenge. Here we describe a large-scale deep recommendation engine that we developed and deployed at Pinterest. We develop a data-efficient Graph Convolutional Network (GCN) algorithm PinSage, which combines efficient random walks and graph convolutions to generate embeddings of nodes (i.e., items) that incorporate both graph structure as well as node feature information. Compared to prior GCN approaches, we develop a novel method based on highly efficient random walks to structure the convolutions and design a novel training strategy that relies on harder-and-harder training examples to improve robustness and convergence of the model. We also develop an efficient MapReduce model inference algorithm to generate embeddings using a trained model. We deploy PinSage at Pinterest and train it on 7.5 billion examples on a graph with 3 billion nodes representing pins and boards, and 18 billion edges. According to offline metrics, user studies and A/B tests, PinSage generates higher-quality recommendations than comparable deep learning and graph-based alternatives. To our knowledge, this is the largest application of deep graph embeddings to date and paves the way for a new generation of web-scale recommender systems based on graph convolutional architectures.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.