Graph Partitioning Research Articles

Combinational Equivalence Checking on AIGs

Combinational Equivalence Checking (CEC) is a critical process in digital circuit design, ensuring that two versions of a circuit are functionally equivalent. Functionally Reduced And-Inverter Graphs (FRAIGs) are a data structure extensively used in CEC, representing Boolean functions as directed acyclic graphs with AND gates and inverters. The main advantage of FRAIGs is their ability to integrate structural hashing with functional reduction, allowing for the elimination of functionally equivalent nodes during graph construction. However, conventional FRAIG approaches face challenges with scalability in complex circuits. To overcome these limitations, we propose three novel methods: improved sampling techniques that refine random simulation and SAT-based methods for early identification of equivalent nodes; advanced graph partitioning strategies that enable parallel processing and localized equivalence checking to accelerate computation; and support node analysis combined with probability distribution modeling to reduce unnecessary checks. Extensive experiments show the effectiveness and efficiency of our proposed methods

A Neural-Network-Based Mapping and Optimization Framework for High-Precision Coarse-Grained Simulation.

The accuracy and efficiency of a coarse-grained (CG) force field are pivotal for high-precision molecular simulations of large systems with complex molecules. We present an automated mapping and optimization framework for molecular simulation (AMOFMS), which is designed to streamline and improve the force field optimization process. It features a neural-network-based mapping function, DSGPM-TP (deep supervised graph partitioning model with type prediction). This model can accurately and efficiently convert atomistic structures to CG mappings, reducing the need for manual intervention. By integrating bottom-up and top-down methodologies, AMOFMS allows users to freely combine these approaches or use them independently as optimization targets. Moreover, users can select and combine different optimizers to meet their specific mission. With its parallel optimizer, AMOFMS significantly accelerates the optimization process, reducing the time required to achieve optimal results. Successful applications of AMOFMS include parameter optimizations for systems such as POPC and PEO, demonstrating its robustness and effectiveness. Overall, AMOFMS provides a general and flexible framework for the automated development of high-precision CG force fields.

Solving the maximum cut problem using Harris Hawk Optimization algorithm.

The objective of the max-cut problem is to cut any graph in such a way that the total weight of the edges that are cut off is maximum in both subsets of vertices that are divided due to the cut of the edges. Although it is an elementary graph partitioning problem, it is one of the most challenging combinatorial optimization-based problems, and tons of application areas make this problem highly admissible. Due to its admissibility, the problem is solved using the Harris Hawk Optimization algorithm (HHO). Though HHO effectively solved some engineering optimization problems, is sensitive to parameter settings and may converge slowly, potentially getting trapped in local optima. Thus, HHO and some additional operators are used to solve the max-cut problem. Crossover and refinement operators are used to modify the fitness of the hawk in such a way that they can provide precise results. A mutation mechanism along with an adjustment operator has improvised the outcome obtained from the updated hawk. To accept the potential result, the acceptance criterion has been used, and then the repair operator is applied in the proposed approach. The proposed system provided comparatively better outcomes on the G-set dataset than other state-of-the-art algorithms. It obtained 533 cuts more than the discrete cuckoo search algorithm in 9 instances, 1036 cuts more than PSO-EDA in 14 instances, and 1021 cuts more than TSHEA in 9 instances. But for four instances, the cuts are lower than PSO-EDA and TSHEA. Besides, the statistical significance has also been tested using the Wilcoxon signed rank test to provide proof of the superior performance of the proposed method. In terms of solution quality, MC-HHO can produce outcomes that are quite competitive when compared to other related state-of-the-art algorithms.

Efficient State Sharding in Blockchain via Density-based Graph Partitioning

Sharding is a promising technique for increasing a blockchain system’s throughput by enabling parallel transaction processing. The main challenge of state sharding lies in ensuring the atomicity verification of cross-sharding transactions, which results in double communication overhead and increases the transaction’s confirmation time. Previous research has primarily focused on developing cross-shard protocols for the fast and reliable validation of transactions involving multiple shards. These studies typically generate a large number of cross-shard transactions because they primarily use simple address mapping for state sharding, that is, the prefix/suffix of the account address. In this article, we propose a state sharding scheme via density-based partitioning of the account-transaction graph. In order to reduce cross-shard transactions, the scheme groups correlated accounts into the same shard by generating the densest subgraphs, as the graph density describes the correlation among accounts, i.e., how often transactions have occurred among accounts. We formulate the graph density-based state sharding problem, with the goal of maximizing the average density across all shards under the workload constraint. We prove the NP-completeness of the problem. To reduce the complexity of finding the densest subgraph, we propose the pruning-based algorithm that reduces the search space by pre-pruning some invalid edges based on the concept of core number. We also extend the linear deterministic greedy algorithm and PageRank algorithm to handle new transactions in the dynamic scenario. We conduct extensive experiments using real transaction data from Ethereum. The experimental results demonstrate a strong correlation between the shard density and the number of cross-shard transactions, and the pruning-based algorithm can reduce the running time by an order of magnitude.

LocalDGP: local degree-balanced graph partitioning for lightweight GNNs

LocalDGP: local degree-balanced graph partitioning for lightweight GNNs

Perfect colourings of hypergraphs

Perfect colourings (equitable partitions) of graphs are extensively studied, while the same concept for hypergraphs attracts much less attention. The aim of this paper is to develop basic notions and properties of perfect colourings for hypergraphs. Firstly, we introduce a multidimensional matrix equation for perfect colourings of hypergraphs and compare this definition with a standard approach based on the incidence graph. Next, we show that the eigenvalues of the parameter matrix of a perfect colouring are eigenvalues of the multidimensional adjacency matrix of a hypergraph. We consider coverings of hypergraphs as a special case of perfect colourings and prove a theorem on the existence of a common covering of two hypergraphs. As an example, we show that a k-transversal in a hypergraph corresponds to a perfect colouring and calculate its parameters. At last, we find all perfect 2-colourings of the Fano's plane hypergraph and compute some eigenvalues of this hypergraph.

Video tracking of single cells to identify clustering behavior

Cancer cell clustering is a critical factor in metastasis, with cells often believed to migrate in groups as they establish themselves in new environments. This study presents preliminary findings from an in vitro experiment, suggesting that co-culturing cells provides an effective method for observing this phenomenon, even though the cells are grown as monolayers. We introduce a novel single-cell tracking approach based on graph theory to identify clusters in PC3 cells cultivated in both monoculture and co-culture with PC12 cells, using 66-h time-lapse recordings. The initial step consists of defining “linked” pairs of PC3 cells, laying the foundation for the application of graph theory. We propose two alternative definitions for cell pairings. The first method, Method 1, defines cells as “linked” at a given time t if they are close together within a defined time period before and after t. A second potential alternative method, Method 2, pairs cells if there is an overlap between the convex hulls of their respective tracks during this time period. Pairing cells enables the application of graph theory for subsequent analysis. This framework represents a cell as a vertex (node) and a relation between two cells as an edge. An interconnected set of high-degree nodes (nodes with many connections or edges) forms a subgraph, or backbone, that defines a patch (cluster) of cells. All nodes connected to this backbone are part of the subgraph. The backbone of high-degree nodes functions as a partition (or cut) of the initial graph. Two consecutive clusters in the video are considered to share the same identity if the following cluster contains at least p = 75 % of the cells from the preceding cluster, and the mean positions of their cells are within △r = 75μm. PC3 cells grown in co-culture appear to form persistent clusters exceeding 10 cells after 40–50 h incubation following seeding. In contrast, PC3 cells cultured alone (mono-culture) did not exhibit this behavior. This approach is experimental and requires further validation with a broader dataset.

Co-Clustering by Directly Solving Bipartite Spectral Graph Partitioning.

Bipartite spectral graph partitioning (BSGP) method as a co-clustering method, has been widely used in document clustering, which simultaneously clusters documents and words by making full use of the duality between documents and words. It consists of two steps: 1) graph construction and 2) singular value decomposition on the bipartite graph to compute a continuous cluster assignment matrix, followed by post-processing to get the discrete solution. However, the generated graph is unstructured and fixed. It heavily relies on the quality of the graph construction. Moreover, the two-stage process may deviate from directly solving the primal problem. In order to tackle these defects, a novel bipartite graph partitioning method is proposed to learn a bipartite graph with exactly c connected components (c is the number of clusters), which can obtain clustering results directly. Furthermore, it is experimentally and theoretically proved that the solution of the proposed model is the discrete solution of the primal BSGP problem for a special situation. Experimental results on synthetic and real-world datasets exhibit the superiority of the proposed method.

DeepMulticut: Deep Learning of Multicut Problem for Neuron Segmentation From Electron Microscopy Volume.

Superpixel aggregation is a powerful tool for automated neuron segmentation from electron microscopy (EM) volume. However, existing graph partitioning methods for superpixel aggregation still involve two separate stages-model estimation and model solving, and therefore model error is inherent. To address this issue, we integrate the two stages and propose an end-to-end aggregation framework based on deep learning of the minimum cost multicut problem called DeepMulticut. The core challenge lies in differentiating the NP-hard multicut problem, whose constraint number is exponential in the problem size. With this in mind, we resort to relaxing the combinatorial solver-the greedy additive edge contraction (GAEC)-to a continuous Soft-GAEC algorithm, whose limit is shown to be the vanilla GAEC. Such relaxation thus allows the DeepMulticut to integrate edge cost estimators, Edge-CNNs, into a differentiable multicut optimization system and allows a decision-oriented loss to feed decision quality back to the Edge-CNNs for adaptive discriminative feature learning. Hence, the model estimators, Edge-CNNs, can be trained to improve partitioning decisions directly while beyond the NP-hardness. Also, we explain the rationale behind the DeepMulticut framework from the perspective of bi-level optimization. Extensive experiments on three public EM datasets demonstrate the effectiveness of the proposed DeepMulticut.

Palm vein template protection scheme for resisting similarity attack

Palm vein recognition technology has rapidly developed due to its high confidentiality and the advantages of liveness detection. Various palm vein template protection methods have emerged to safeguard palm vein data from theft and attack. To fulfill the performance loss requirement of the ideal biometric template protection scheme, these methods make the distribution of the original palm vein data and the palm vein protection templates have a strong distance preserving property (similarity preserving), making it difficult to defend against similarity attack (SA). To address these risks and prevent palm vein data leakage, we propose a Nonlinear Spectral Hashing (NSH) method for palm vein template protection. To obtain palm vein templates with both performance and security, the method first performs random projection on palm vein data to obtain revocable and unlinkable palm vein features. Subsequently, through spectral graph partitioning, it achieves mapping with a preserved similarity structure for palm veins, avoiding excessive performance loss. The method then employs a nonlinear activation function to alter the distribution of palm vein templates with large post-mapping differences, resulting in a uniform distribution of inter-class distances for palm vein data. By reducing the distance-preserving properties, the method enhances protection against similarity attacks. Finally, a sign function is applied to obtain non-invertible binary palm vein templates. Experimental evaluations on the public Tongji University palm vein database assess the method's non-invertibility, revocability, unlinkability, resistance to similarity attack, and recognition performance. The results indicate an Equal Error Rate (EER) of 0.50 %, demonstrating the method's ability to maintain good recognition performance while ensuring high security.

Dual Clustering-Based Method for Geospatial Knowledge Graph Partitioning

Geospatial knowledge graphs provide critical technology for integrating geographic information and semantic knowledge, which are very useful for geographic data analysis. As the scale of geospatial knowledge graphs continues to grow, the distributed management of geospatial knowledge graphs is becoming an inevitable requirement. Geospatial knowledge graph partitioning is the core technology for the distributed management of geospatial knowledge graphs. To support geographic data analysis, spatial relationships between entities should be considered in the application of geospatial knowledge graphs. However, existing knowledge graph partitioning methods overlook the spatial relationships between entities, resulting in the low efficiency of spatial queries. To address this issue, this study proposes a geospatial knowledge graph partitioning method based on dual clustering which performs two different clustering methods step by step. First, the density peak clustering method (DPC) is used to cluster geographic nodes. The nodes within each cluster are merged into a super-node. Then, we use an efficient graph clustering method (i.e., Leiden) to identify the community structure of the graph. Nodes belonging to the same community are further merged to reduce the size of the graph. Finally, partitioning operations are performed on the compressed graph based on the idea of the Linear-Weighted Deterministic Greedy Policy (LDG). We construct a geospatial knowledge graph based on YAGO3 to evaluate the performance of the proposed graph partitioning method. The experimental results show that the proposed method outperforms ten comparison methods in terms of graph partitioning quality and spatial query efficiency.

A multi-objective partitioning algorithm for large-scale graph based on NSGA-II

Graph partitioning plays a crucial role in distributed graph computation. The existing algorithms often formulate the partition problem as a single objective optimization problem focusing on load balancing and minimizing communication overhead, which is difficult to achieve optimization for multiple objectives and neglects communication balance. This paper formulates the problem of graph partitioning as a multi-objective constrained optimization problem, and proposes an enhanced NSGA-II to solve it. The objectives include achieving load balance, minimizing the communication overhead, and keeping communication balance. After partitioning the graph according to the vertices’ degrees in first, the NSGA-II combined neighbor search is applied to guide the vertices migration to obtain the best partition result. Compared with Hash, GAP, HLP, CP_WSG, and SPNL using dataset from SNAP, the experiments performed on a cluster of 4 computing nodes demonstrate that the load balance ratio and cut-edge ratio of the proposed algorithm is almost the same with those of the best existing algorithm, and it can achieve the better cut-edge balance ratio and communication balance ratio than the other algorithms.

Near-Data Source Graph Partitioning

Recently, numerous graph partitioning approaches have been proposed to distribute a big graph to machines in a cluster for distributed computing. Due to heavy communication overhead, these graph partitioning approaches always suffered from long ingress times. Also, heavy communication overhead not only limits the scalability of distributed graph-parallel computing platforms but also reduces the overall performance of clusters. In order to address this problem, this work proposed a near-data source parallel graph partitioning approach noted as NDGP. In NDGP, an edge was preferentially distributed to the machine where it was stored. We implemented NDGP over two classic graph partitioning approaches, Random and Greedy, and one most recently proposed graph partitioning approach, OLPGP, and evaluated its effectiveness. Extensive experiments conducted on real-world data sets verified the effectiveness of NDGP on reducing the communication overhead in the graph partitioning process and demonstrated that NDGP does not induce additional communication and computing workload to the graph-distributed computing that follows.

Robust underwater SLAM fusing bathymetric and range information

Bathymetric simultaneous localization and mapping (BSLAM) can accurately locate an autonomous underwater vehicle (AUV) in deep-sea missions, but it cannot provide real-time location results for vehicles prior to revisit due to the lack of overlap between bathymetric measurements. A real-time estimate of the vehicle’s location can not only help the vehicle to follow the given path accurately but reduce the number of invalid bathymetric associations caused by the nearly flat seafloor terrain. In this paper, we proposed a bathymetric and range information fused underwater SLAM (BRSLAM) method that estimates the vehicle’s position in real time by combining both bathymetric data and range measurements. In particular, a probabilistic graph partitioning algorithm (PGPA) was proposed to identify range outliers, and an information-theoretic node reduction method was presented to reduce the computational complexity of BRSLAM. Playback experiments showed that the BRSLAM can provide accurate navigation results with no more than 4 beacons.

Partitioning problems via random processes

AbstractThere are a number of well‐known problems and conjectures about partitioning graphs to satisfy local constraints. For example, the majority colouring conjecture of Kreutzer, Oum, Seymour, van der Zypen and Wood states that every directed graph has a 3‐colouring such that for every vertex , at most half of the out‐neighbours of have the same colour as . As another example, the internal partition conjecture, due to DeVos and to Ban and Linial, states that for every , all but finitely many ‐regular graphs have a partition into two non‐empty parts such that for every vertex , at least half of the neighbours of lie in the same part as . We prove several results in this spirit: in particular, two of our results are that the majority colouring conjecture holds for Erdős–Rényi random directed graphs (of any density), and that the internal partition conjecture holds if we permit a tiny number of ‘exceptional vertices’. Our proofs involve a variety of techniques, including several different methods to analyse random recolouring processes. One highlight is a personality‐changing scheme: we ‘forget’ certain information based on the state of a Markov chain, giving us more independence to work with.

Bad Local Minima Exist in the Stochastic Block Model

We study the disassortative stochastic block model with three communities, a well-studied model of graph partitioning and Bayesian inference for which detailed predictions based on the cavity method exist (Decelle et al. in Phys Rev E 84:066106, 2011). We provide strong evidence that for a part of the phase where efficient algorithms exist that approximately reconstruct the communities, inference based on maximum a posteriori (MAP) fails. In other words, we show that there exist modes of the posterior distribution that have a vanishing agreement with the ground truth. The proof is based on the analysis of a graph colouring algorithm from Achlioptas and Moore (J Comput Syst Sci 67:441–471, 2003).

Optimizing Connected Components Graph Partitioning With Minimum Size Constraints Using Integer Programming and Spectral Clustering Techniques

ABSTRACTIn this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with minimum size, where each subset induces a connected subgraph with minimal edge cost. The problem naturally surges in applications where connectivity is essential, such as cluster detection in social networks, political districting, sports team realignment, and energy distribution. Mixed Integer Programming formulations together with a variety of valid inequalities are demonstrated and computationally tested. An assisted column generation approach by spectral clustering is also proposed for this problem with additional valid inequalities. Finally, the methods are tested for several simulated instances, and computational results are discussed. Overall, the proposed column generation technique enhanced by spectral clustering offers a promising approach to solve clustering and partitioning problems.

On perfectly friendly bisections of random graphs

On perfectly friendly bisections of random graphs

Maximum Bisections of Graphs with Girth at Least Six

Maximum Bisections of Graphs with Girth at Least Six

An end-to-end bi-objective approach to deep graph partitioning

Graphs are ubiquitous in real-world applications, such as computation graphs and social networks. Partitioning large graphs into smaller, balanced partitions is often essential, with the bi-objective graph partitioning problem aiming to minimize both the “cut” across partitions and the imbalance in partition sizes. However, existing heuristic methods face scalability challenges or overlook partition balance, leading to suboptimal results. Recent deep learning approaches, while promising, typically focus only on node-level features and lack a truly end-to-end framework, resulting in limited performance. In this paper, we introduce a novel method based on graph neural networks (GNNs) that leverages multilevel graph features and addresses the problem end-to-end through a bi-objective formulation. Our approach explores node-, local-, and global-level features, and introduces a well-bounded bi-objective function that minimizes the cut while ensuring partition-wise balance across all partitions. Additionally, we propose a GNN-based deep model incorporating a Hardmax operator, allowing the model to optimize partitions in a fully end-to-end manner. Experimental results on 12 datasets across various applications and scales demonstrate that our method significantly improves both partitioning quality and scalability compared to existing bi-objective and deep graph partitioning baselines.