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Given an undirected graph $G$ and $q$ integers $n_1,n_2,n_3, \cdots, n_q$, balanced connected $q$-partition problem ($BCP_q$) asks whether there exists a partition of the vertex set $V$ of $G$ into $q$ parts $V_1,V_2,V_3,\cdots, V_q$ such that for all $i\in[1,q]$, $|V_i|=n_i$ and the graph induced on $V_i$ is connected. A related problem denoted as the balanced connected $q$-edge partition problem ($BCEP_q$) is defined as follows. Given an undirected graph $G$ and $q$ integers $n_1,n_2,n_3, \cdots, n_q$, $BCEP_q$ asks whether there exists a partition of the edge set of $G$ into $q$ parts $E_1,E_2,E_3,\cdots, E_q$ such that for all $i\in[1,q]$, $|E_i|=n_i$ and the graph induced on the edge set $E_i$ is connected. Here we study both the problems for $q=2$ and prove that $BCP_q$ for $q\geq 2$ is $W[1]$-hard. We also show that $BCP_2$ is unlikely to have a polynomial kernel on the class of planar graphs. Coming to the positive results, we show that $BCP_2$ is fixed parameter tractable (FPT) parameterized by treewidth of the graph, which generalizes to FPT algorithm for planar graphs. We design another FPT algorithm and a polynomial kernel on the class of unit disk graphs parameterized by $\min(n_1,n_2)$. Finally, we prove that unlike $BCP_2$, $BCEP_2$ is FPT parameterized by $\min(n_1,n_2)$.

Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ starts from $s_i$ and ends at $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. We also classify the complexity of the second problem (subject to one missing case) if the number of terminal sets is fixed, that is, not part of the input.

Graph neural networks (GNNs) are emerging machine learning models on graphs. Permutation-equivariance and proximity-awareness are two important properties highly desirable for GNNs. Both properties are needed to tackle some challenging graph problems, such as finding communities and leaders. In this paper, we first analytically show that the existing GNNs, mostly based on the message-passing mechanism, cannot simultaneously preserve the two properties. Then, we propose Stochastic Message Passing (SMP) model, a general and simple GNN to maintain both proximity-awareness and permutation-equivariance. In order to preserve node proximities, we augment the existing GNNs with stochastic node representations. We theoretically prove that the mechanism can enable GNNs to preserve node proximities, and at the same time, maintain permutation-equivariance with certain parametrization. We report extensive experimental results on ten datasets and demonstrate the effectiveness and efficiency of SMP for various typical graph mining tasks, including graph reconstruction, node classification, and link prediction.

Theorem proving is one of the oldest applications which require heuristics to prune the search space. Invertible proof procedures has been the major tool. In this paper, we present a novel and powerful heuristic called $nongshim$ which can be seen as an underlying principle of invertible proof procedures. Using this heuristic, we derive an invertible sequent calculus\cite{Ketonen,Troe} from sequent calculus for propositional logic.

A famous result by Erd\H{o}s and Szekeres (1935) asserts that, for all $k,d \in \mathbb{N}$, there is a smallest integer $n = g^{(d)}(k)$ such that every set of at least $n$ points in $\mathbb{R}^d$ in general position contains a $k$-gon, that is, a subset of $k$ points which is in convex position. In this article, we present a SAT model based on acyclic chirotopes (oriented matroids) to investigate Erd\H{o}s--Szekeres numbers in small dimensions. To solve the SAT instances we use modern SAT solvers and all our unsatisfiability results are verified using DRAT certificates. We show $g^{(3)}(7) = 13$, $g^{(4)}(8) \le 13$, and $g^{(5)}(9) \le 13$, which are the first improvements for decades. For the setting of $k$-holes (i.e., $k$-gons with no other points in the convex hull), where $h^{(d)}(k)$ denotes the minimum number $n$ such that every set of at least $n$ points in $\mathbb{R}^d$ in general position contains a $k$-hole, we show $h^{(3)}(7) \le 14$, $h^{(4)}(8) \le 13$, and $h^{(5)}(9) \le 13$. Moreover, all obtained bounds are sharp in the setting of acyclic chirotopes and we conjecture them to be sharp also in the original setting of point sets. As a byproduct, we verify previously known bounds. In particular, we present the first computer-assisted proof of the upper bound $h^{(2)}(6)\le g^{(2)}(9) \le 1717$ by Gerken (2008).

Graph Neural Networks (GNNs) have recently become increasingly popular due to their ability to learn complex systems of relations or interactions arising in a broad spectrum of problems ranging from biology and particle physics to social networks and recommendation systems. Despite the plethora of different models for deep learning on graphs, few approaches have been proposed thus far for dealing with graphs that present some sort of dynamic nature (e.g. evolving features or connectivity over time). In this paper, we present Temporal Graph Networks (TGNs), a generic, efficient framework for deep learning on dynamic graphs represented as sequences of timed events. Thanks to a novel combination of memory modules and graph-based operators, TGNs are able to significantly outperform previous approaches being at the same time more computationally efficient. We furthermore show that several previous models for learning on dynamic graphs can be cast as specific instances of our framework. We perform a detailed ablation study of different components of our framework and devise the best configuration that achieves state-of-the-art performance on several transductive and inductive prediction tasks for dynamic graphs.

Learning structural representations of node sets from graph-structured data is crucial for applications ranging from node-role discovery to link prediction and molecule classification. Graph Neural Networks (GNNs) have achieved great success in structural representation learning. However, most GNNs are limited by the 1-Weisfeiler-Lehman (WL) test and thus possible to generate identical representation for structures and graphs that are actually different. More powerful GNNs, proposed recently by mimicking higher-order-WL tests, only focus on entire-graph representations and cannot utilize sparsity of the graph structure to be computationally efficient. Here we propose a general class of structure-related features, termed Distance Encoding (DE), to assist GNNs in representing node sets with arbitrary sizes with strictly more expressive power than the 1-WL test. DE essentially captures the distance between the node set whose representation is to be learnt and each node in the graph, which includes important graph-related measures such as shortest-path-distance and generalized PageRank scores. We propose two general frameworks for GNNs to use DEs (1) as extra node attributes and (2) further as controllers of message aggregation in GNNs. Both frameworks may still utilize the sparse structure to keep scalability to process large graphs. In theory, we prove that these two frameworks can distinguish node sets embedded in almost all regular graphs where traditional GNNs always fail. We also rigorously analyze their limitations. Empirically, we evaluate these two frameworks on node structural roles prediction, link prediction and triangle prediction over six real networks. The results show that our models outperform GNNs without DEs by up-to 15% improvement in average accuracy and AUC. Our models also significantly outperform other SOTA baselines particularly designed for those tasks.

Graph neural networks (GNNs) are typically applied to static graphs that are assumed to be known upfront. This static input structure is often informed purely by insight of the machine learning practitioner, and might not be optimal for the actual task the GNN is solving. In absence of reliable domain expertise, one might resort to inferring the latent graph structure, which is often difficult due to the vast search space of possible graphs. Here we introduce Pointer Graph Networks (PGNs) which augment sets or graphs with additional inferred edges for improved model expressivity. PGNs allow each node to dynamically point to another node, followed by message passing over these pointers. The sparsity of this adaptable graph structure makes learning tractable while still being sufficiently expressive to simulate complex algorithms. Critically, the pointing mechanism is directly supervised to model long-term sequences of operations on classical data structures, incorporating useful structural inductive biases from theoretical computer science. Qualitatively, we demonstrate that PGNs can learn parallelisable variants of pointer-based data structures, namely disjoint set unions and link/cut trees. PGNs generalise out-of-distribution to 5x larger test inputs on dynamic graph connectivity tasks, outperforming unrestricted GNNs and Deep Sets.

Learning low-dimensional embeddings of knowledge graphs is a powerful approach used to predict unobserved or missing edges between entities. However, an open challenge in this area is developing techniques that can go beyond simple edge prediction and handle more complex logical queries, which might involve multiple unobserved edges, entities, and variables. For instance, given an incomplete biological knowledge graph, we might want to predict "em what drugs are likely to target proteins involved with both diseases X and Y?" -- a query that requires reasoning about all possible proteins that {\em might} interact with diseases X and Y. Here we introduce a framework to efficiently make predictions about conjunctive logical queries -- a flexible but tractable subset of first-order logic -- on incomplete knowledge graphs. In our approach, we embed graph nodes in a low-dimensional space and represent logical operators as learned geometric operations (e.g., translation, rotation) in this embedding space. By performing logical operations within a low-dimensional embedding space, our approach achieves a time complexity that is linear in the number of query variables, compared to the exponential complexity required by a naive enumeration-based approach. We demonstrate the utility of this framework in two application studies on real-world datasets with millions of relations: predicting logical relationships in a network of drug-gene-disease interactions and in a graph-based representation of social interactions derived from a popular web forum.

Neural networks can learn to represent and manipulate numerical information, but they seldom generalize well outside of the range of numerical values encountered during training. To encourage more systematic numerical extrapolation, we propose an architecture that represents numerical quantities as linear activations which are manipulated using primitive arithmetic operators, controlled by learned gates. We call this module a neural arithmetic logic unit (NALU), by analogy to the arithmetic logic unit in traditional processors. Experiments show that NALU-enhanced neural networks can learn to track time, perform arithmetic over images of numbers, translate numerical language into real-valued scalars, execute computer code, and count objects in images. In contrast to conventional architectures, we obtain substantially better generalization both inside and outside of the range of numerical values encountered during training, often extrapolating orders of magnitude beyond trained numerical ranges.