Graph Partitioning Research Articles

HubHSP graph: Capturing local geometrical and statistical data properties via spanning graphs

The computation of a continuous generative model to describe a finite sample of an infinite metric space can prove challenging and lead to erroneous hypothesis, particularly in high-dimensional spaces. In this paper, we follow a different route and define the Hubness Half Space Partitioning graph (HubHSP graph). By constructing this spanning graph over the dataset, we can capture both the geometrical and statistical properties of the data without resorting to any continuity assumption. Leveraging the classical graph-theoretic apparatus, the HubHSP graph facilitates critical operations, including the creation of a representative sample of the original dataset, without relying on density estimation. This representative subsample is essential for a range of operations, including indexing, visualization, and machine learning tasks such as clustering or inductive learning. With the HubHSP graph, we can bypass the limitations of traditional methods and obtain a holistic understanding of our dataset’s properties, enabling us to unlock its full potential.

Some Exact Results for Non-Degenerate Generalized Turán Problems

The generalized Turán number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $\chi(H)<\chi(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when the extremal graph is a complete $(\chi(F)-1)$-partite graph. We obtain multiple exact results with other kinds of extremal graphs.

Labeled graph partitioning scheme for distributed edge caching

Distributed edge caching could address latency and congestion problems in large-scale data access effectively, improving system throughput and performance. However, the lack of specialized edge caching solutions for graph data resulted in low cache hit rates and high data access latency. This limitation stemmed from existing graph partitioning schemes that overlooked the specific requirements of edge caching, such as communication overhead between edge servers, data query frequency, and query patterns. To overcome these challenges, we proposed LGPE, a labeled graph partitioning scheme for distributed edge caching based on frequent query patterns. LGPE generated frequent query patterns according to the user’s historical query subgraphs and then performed labeled graph partitioning based on the frequent pattern, ensuring that the labeled graph divided into edge servers had a high hit rate for frequent user queries while satisfying a minimum edge cut. Evaluation on real datasets from various application domains demonstrated that LGPE achieved approximately 40% higher cache hit rates compared to benchmarks like METIS and SCOTCH, and it also performed well in terms of edge cuts in special graph cases.

DGC: Training Dynamic Graphs with Spatio-Temporal Non-Uniformity using Graph Partitioning by Chunks

Dynamic Graph Neural Network (DGNN) has shown a strong capability of learning dynamic graphs by exploiting both spatial and temporal features. Although DGNN has recently received considerable attention by AI community and various DGNN models have been proposed, building a distributed system for efficient DGNN training is still challenging. It has been well recognized that how to partition the dynamic graph and assign workloads to multiple GPUs plays a critical role in training acceleration. Existing works partition a dynamic graph into snapshots or temporal sequences, which only work well when the graph has uniform spatio-temporal structures. However, dynamic graphs in practice are not uniformly structured, with some snapshots being very dense while others are sparse. To address this issue, we propose DGC, a distributed DGNN training system that achieves a 1.25× - 7.52× speedup over the state-of-the-art in our testbed. DGC's success stems from a new graph partitioning method that partitions dynamic graphs into chunks, which are essentially subgraphs with modest training workloads and few inter connections. This partitioning algorithm is based on graph coarsening, which can run very fast on large graphs. In addition, DGC has a highly efficient run-time, powered by the proposed chunk fusion and adaptive stale aggregation techniques. Extensive experimental results on 3 typical DGNN models and 4 popular dynamic graph datasets are presented to show the effectiveness of DGC.

Graph partitions under average degree constraint

In this paper, we prove that every graph with average degree at least s+t+2 has a vertex partition into two parts, such that one part has average degree at least s, and the other part has average degree at least t. This solves a conjecture of Csóka, Lo, Norin, Wu and Yepremyan.

L-BGNN: Layerwise Trained Bipartite Graph Neural Networks.

Learning low-dimensional representations of bipartite graphs enables e-commerce applications, such as recommendation, classification, and link prediction. A layerwise-trained bipartite graph neural network (L-BGNN) embedding method, which is unsupervised, efficient, and scalable, is proposed in this work. To aggregate the information across and within two partitions of a bipartite graph, a customized interdomain message passing (IDMP) operation and an intradomain alignment (IDA) operation are adopted by the proposed L-BGNN method. Furthermore, we develop a layerwise training algorithm for L-BGNN to capture the multihop relationship of large bipartite networks and improve training efficiency. We conduct extensive experiments on several datasets and downstream tasks of various scales to demonstrate the effectiveness and efficiency of the L-BGNN method as compared with state-of-the-art methods. Our codes are publicly available at https://github.com/TianXieUSC/L-BGNN.

Approximation Algorithms for Graph Partition into Bounded Independent Sets

Approximation Algorithms for Graph Partition into Bounded Independent Sets

An approximation algorithm for high-dimensional table compression on balanced [formula omitted]-partite graph

This paper considers the high-dimensional table compression problem on balanced k-partite graph. The objective is to choose half of the vertices from each of the k-partite to maximize the total weight of edges connecting the chosen vertices. Our main contribution is a 0.8785-approximation algorithm for the high-dimensional table compression problem by introducing the α-independent solutions through Lasserre semidefinite programming. This new algorithm improved two previous low-dimensional results, namely the 0.8731-approximation algorithm of Wu et al. for the one-dimensional case and the 0.6708-approximation of Xu and Du for the two-dimensional case.

Agglomeration of polygonal grids using graph neural networks with applications to multigrid solvers

Agglomeration-based strategies are important both within adaptive refinement algorithms and to construct scalable multilevel algebraic solvers. In order to automatically perform agglomeration of polygonal grids, we propose the use of Machine Learning (ML) strategies, that can naturally exploit geometrical information about the mesh in order to preserve the grid quality, enhancing performance of numerical methods and reducing the overall computational cost. In particular, we employ the k-means clustering algorithm and Graph Neural Networks (GNNs) to partition the connectivity graph of a computational mesh. Moreover, GNNs have high online inference speed and the advantage to process naturally and simultaneously both the graph structure of mesh and the geometrical information, such as the areas of the elements or their barycentric coordinates. These techniques are compared with METIS, a standard algorithm for graph partitioning, which is meant to process only the graph information of the mesh. We demonstrate that performance in terms of quality metrics is enhanced for ML strategies. Such models also show a good degree of generalization when applied to more complex geometries, such as brain MRI scans, and the capability of preserving the quality of the grid. The effectiveness of these strategies is demonstrated also when applied to MultiGrid (MG) solvers in a Polygonal Discontinuous Galerkin (PolyDG) framework. In the considered experiments, GNNs show overall the best performance in terms of inference speed, accuracy and flexibility of the approach.

Unfriendly partitions when avoiding vertices of finite degree

Abstract An unfriendly partition of a graph $G = (V,E)$ is a function $c: V \to 2$ such that $|\{x\in N(v): c(x)\neq c(v)\}|\geq |\{x\in N(v): c(x)=c(v)\}|$ for every vertex $v\in V$, where $N(v)$ denotes its neighborhood. It was conjectured by Cowan and Emerson [2] that every graph has an unfriendly partition, but Milner and Shelah in [5] found counterexamples for that statement by analysing graphs with uncountably many vertices. Curiously, none of their graphs have vertices with finite degree. Therefore, as a natural direction to approach, in this paper we search for the least cardinality of a graph with this property and that admits no unfriendly partitions. Actually, among some other independence results, we conclude that this value cannot be precisely determined within $\textrm {ZFC}$, in the sense that it may vary from model to model of set theory.

Blockchain Exchange Optimization by Graph Partition and Merging in IoT Scenarios

Blockchain exchanges have been widely used for IoT-based resource sharing. An exchange consists of several related transactions where participants exchange their assets. However, the exchange can take a long time to complete due to different reasons (such as IoT device failure). Meanwhile, exchange fees are an important aspect, because an exchange usually contains multiple transactions, and the exchange fee is the sum of the fees for each transaction. In this paper, we explore ways to optimize the exchange process concerning the above issues. The method is based on the graph of an exchange, which is strongly connected. To save exchange time, we propose a way to split the exchange into smaller exchanges, since smaller exchanges have fewer transactions. The method first finds strongly connected sub graphs, separates them from the original graph, and adjusts the edges and their weights. To save transaction fees, we propose a method to reduce the number of transactions. Transactions that meet certain requirements are combined into one transaction, and corresponding weights are adjusted. The verification results show that the proposed method can optimize the exchange process, which can be used to save the exchange time and also can be used to save transaction fees.

Controlled Remote Implementation of Operations via Graph States

AbstractProtocols are proposed for controlled remote implementation of operations here. Sharing a ‐partite graph state, 2N participants collaborate to prepare the stator and realize the operation on N unknown states for distributed systems , only with the permission of a controller. The control power analysis shows that without the controller's permission, eight specified operations can be implemented with the success rate of 50%, other operations can be realized with the success rate of 25%, and thus the control power is reliable. All the implementation requirements of this protocol can be satisfied by means of local operations and classical communications, and the experimental feasibility is presented according to current techniques. The entanglement requirement of this protocol is characterized in terms of geometric measure of entanglement. It turns out to be economic to realize the control function from the perspective of entanglement cost.

Incomplete multi-view clustering via structure exploration and missing-view inference

Incomplete multi-view clustering (IMVC) aims to boost clustering performance by capturing complementary information from incomplete multi-views, where partial data samples in one or more views are missing. Current IMVC methods mostly impute missing samples at the guidance of the global/local structure or directly learn a common representation without imputation using subspace or graph learning techniques. However, the consistent and inconsistent structures across views are often ignored during imputation, leading to the introduction of noise and biases. Additionally, lacking the handling of missing samples would mislead the learning methods and degrade clustering performance. To this end, we propose a novel approach called Structure Exploration and Missing-view Inference (SEMI) for IMVC. Specifically, SEMI explores the underlying multi-structures of data, including global, local, consistent, and inconsistent structures, by jointly modeling self-expression subspace, graph, and clustering-oriented partition learning. This enables the capture of consistent and discriminative information and fuses it into a unified coefficient matrix. The learned coefficient matrix with the explored multi-structures then guides the inference of missing views, facilitating the alleviation of the influence of existing noise and biases and the mitigation of the introduction of further noise and biases. These two components are seamlessly integrated and mutually improved through an efficient alternating optimization strategy. Experimental results demonstrate the effectiveness and superior performance of the proposed method.

Generation of Dissimilar Alternative Product Formulations Using Graphs

In this work, alternative product formulations are represented as a bipartite graph, and mixed-integer programming is used to find a graph partitioning that identifies the most dissimilar formulations within a potentially large design space. This involves fully integrating available models for product properties and known heuristic rules for product formulation. The set of dissimilar alternative formulations thus found constitutes the best exploratory plan of experiments in the face of the available knowledge. Also in this work, a cosmetic emulsion example is explored, where some ingredients (in a non-specified number) have to be chosen from a pool of 32 possible. The number of possible combinations of ingredients is around half a million. With the new tools proposed herein, one is able to identify small sets of alternative formulations that adhere to available models and known heuristics rules, have maximum dissimilarity, and are optimal regarding a specific product design objective (cost or other), in a few minutes of computational time.

FTW-GAT: An FPGA-Based Accelerator for Graph Attention Networks With Ternary Weights

Graph attention networks (GATs) are a mainstream graph neural network (GNN) model. They have better performance in some tasks compared to other GNN models. The challenge is that graph data structures are irregular and data dependency in GATs is complex. General-purpose hardware cannot provide enough performance or energy efficiency. Therefore, a specialized accelerator for GATs is needed. In this brief, we propose FTW-GAT to accelerate GAT inference. The key idea of our approach is to quantize the weights of GATs to ternary values, which can greatly simplify processing elements (PE), eliminate the dependence on digital signal processors (DSPs) and reduce power consumption. Then, we use operation fusion, multi-level pipelining and graph partitioning to improve the parallelism. Finally, we implement the accelerator on a Xilinx VCU128 FPGA platform. The results show that FTW-GAT achieves performance speedup by 390×, 17× and 1.4×, and energy efficiency by 4007×, 261× and 3.1× compared to CPUs, GPUs and a prior GAT accelerator.

Object Affinity Learning: Towards Annotation-Free Instance Segmentation.

We address the problem of annotation-free instance segmentation in the wild, aiming to relieve the expensive cost of manual mask annotations. Existing approaches utilize appearance cues, such as color, edge, and texture information, to generate pseudo masks for instance segmentation. However, due to the ambiguity of defining an object by visual appearance alone, these methods fail to distinguish objects from the background under complex scenes. Beyond visual cues, objects are one-piece in space and move together over time, which indicates that geometry cues, such as spatial continuity and motion consistency, are also exploitable for this problem. To directly utilize geometry cues, we propose an affinity-based paradigm for annotation-free instance segmentation. The new paradigm is called object affinity learning, a proxy task of annotation-free instance segmentation, which aims to tell whether two pixels come from the same object by learning feature representation from geometry cues. During inference, the learned object affinity could be further converted into instance segmentation masks by some graph partition algorithms. The proposed object affinity learning achieves much better instance segmentation performance than existing pseudo-mask-based methods on the large-scale Waymo Open Dataset and KITTI dataset.

SUGAR: Efficient Subgraph-Level Training via Resource-Aware Graph Partitioning

SUGAR: Efficient Subgraph-Level Training via Resource-Aware Graph Partitioning

Graph Partitions in Chemistry.

We study partitions (equitable, externally equitable, or other) of graphs that describe physico-chemical systems at the atomic or molecular level; provide examples that show how these partitions are intimately related with symmetries of the systems; and discuss how such a link can further lead to insightful relations with the systems' physical and chemical properties. We define a particular kind of graph partition, which we call Chemical Equitable Partition (CEP), accounting for chemical composition as well as connectivity and associate it with a quantitative measure of information reduction that accompanies its derivation. These concepts are applied to model molecular and crystalline solid systems, illustrating their potential as a means to classify atoms according to their chemical or crystallographic role. We also cluster materials in meaningful manners that take their microstructure into account and even correlate them with the materials' physical properties.

There is no going back: Properties of the non-backtracking Laplacian

We prove new properties of the non-backtracking graph and the non-backtracking Laplacian for graphs. In particular, among other results, we prove that two simple graphs are isomorphic if and only if their corresponding non-backtracking graphs are isomorphic, and we investigate properties of various classes of non-backtracking Laplacian eigenfunctions, such as symmetric and antisymmetric eigenfunctions. Moreover, we introduce and study circularly partite graphs as a generalization of bipartite graphs, and we use this notion to state a sharp upper bound for the spectral gap from 1. We also investigate the singular values of the non-backtracking Laplacian in relation to independence numbers, and we use them to bound the moduli of the eigenvalues.

Linear-size formulations for connected planar graph partitioning and political districting

Linear-size formulations for connected planar graph partitioning and political districting