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With the rapid growth of real-world graphs, the size of which can easily exceed the on-chip (board) storage capacity of an accelerator, processing large-scale graphs on a single Field Programmable Gate Array (FPGA) becomes difficult. The multi-FPGA acceleration is of great necessity and importance. Many cloud providers (e.g., Amazon, Microsoft, and Baidu) now expose FPGAs to users in their data centers, providing opportunities to accelerate large-scale graph processing. In this paper, we present a communication library, called FDGLib, which can easily scale out any existing single FPGA-based graph accelerator to a distributed version in a data center, with minimal hardware engineering efforts. FDGLib provides six APIs that can be easily used and integrated into any FPGA-based graph accelerator with only a few lines of code modifications. Considering the torus-based FPGA interconnection in data centers, FDGLib also improves communication efficiency using simple yet effective torus-friendly graph partition and placement schemes. We interface FDGLib into AccuGraph, a state-of-the-art graph accelerator. Our results on a 32-node Microsoft Catapult-like data center show that the distributed AccuGraph can be 2.32x and 4.77x faster than a state-of-the-art distributed FPGA-based graph accelerator ForeGraph and a distributed CPU-based graph system Gemini, with better scalability.

A domain of problem-solving models the problems using graphs, for the graphs are effective representation of such problems, leading to their efficient solutions. The nodes in a graph represent a division of unit work—the computation, and the connecting edges represent communication required among the nodes to accomplish that unit work. The weight is assigned to the nodes and connecting edges for the cost incurred to compute and to collaborate, respectively. Graph partitioning exploits the concurrency in the problem being modeled and maps the problem onto parallel processors to guarantee efficient and load-balanced execution. The objective is to—(i) equally distribute the computations on available computing power (parallel processors) and (ii) minimize the cost of collaboration. To achieve the said objectives for any complex problem, the spectral graph partitioning is demonstrated here—that uses eigenvectors of the graph’s laplacian matrix. The results are tested via the realization of the stochastic block model. The quality of graph partitioning is tested by comparing it with ground truth results. Further, for a large-scale graph, the parallel implementation of spectral graph partitioning on GPGPU is presented. The GPGPU implementation provides better speedup with scalability.

The stability of a social network has been widely studied as an important indicator for both the network holders and the participants. Existing works on reinforcing networks focus on a local view, e.g., the anchored $$k$$ -core problem aims to enlarge the size of the $$k$$ -core with a fixed input k. Nevertheless, it is more promising to reinforce a social network in a global manner: considering the engagement of every user (vertex) in the network. Since the coreness of a user has been validated as the “best practice” for capturing user engagement, we propose and study the anchored coreness problem in this paper: anchoring a small number of vertices to maximize the coreness gain (the total increment of coreness) of all the vertices in the network. We prove the problem is NP-hard and show it is more challenging than the existing local-view problems. An efficient greedy algorithm is proposed with novel techniques on pruning search space and reusing the intermediate results. The algorithm is also extended to distributed environment with a novel graph partition strategy to ensure the computing independency of each machine. Extensive experiments on real-life data demonstrate that our model is effective for reinforcing social networks and our algorithms are efficient.

We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic multi-level nature of a DFN. DFN models are used to simulate flow and transport through low-permeability fractured media in the subsurface by explicitly representing fractures as discrete entities. The governing equations for flow and transport are numerically integrated on computational meshes generated on the interconnected fracture networks. Modern high-fidelity DFN simulations require high-performance computing on multiple processors where performance and scalability depends partially on obtaining a high-quality partition of the mesh to balance work-loads and minimize communication across all processors. The discrete structure of a DFN naturally lends itself to various graph representations, which can be thought of as coarse-scale representations of the computational mesh. Using this concept, we develop two applications of the multilevel graph partitioning algorithm to partition the mesh of a DFN. In the first, we project a partition of the graph based on the DFN topology onto the mesh of the DFN and in the second, this DFN-based projection is used as the initial condition for further partitioning refinement of the mesh. We compare the performance of these methods with standard multi-level graph partitioning using graph-based metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. The DFN-based and the mesh-based partitioning methods are comparable in terms of the graph-based metrics, but the time required to obtain the partition is several orders of magnitude faster using the DFN-based partitions. The computation-based metrics show comparable performance between both methods so, in combination, the DFN-based partitions are several orders of magnitude faster than the mesh-based partition. Moreover, the method which uses the DFN-partition solution as the initial condition of the mesh partition provided cut and imbalance values that were close to the mesh-based partition but in a fraction of the time. In turn, this hybrid method outperformed both of the other methods in terms of the total run time.

With a profound effect of online reviews on customers’ decisions about purchasing products or services, untruthful (fake) reviews written to deceive product quality and receive unfair commercial benefits have become a crucial problem. In this work, we propose a graph partitioning approach (BeGP) and its extension (BeGPX) to distinguish fake reviewers from benign ones. The main idea of BeGP is to first construct a behavioral graph in which reviewers are connected if they share common characteristic features that capture their similar behavior. Then, the algorithm starts with a small subgraph of known fake reviewers and afterwards repeatedly expands the subgraph by inducing other connected suspicious reviewers. Subsequently, all reviews of those suspects are hypothesized to be untruthful. Moreover, to enhance the performance of fake review(er) detection, BeGPX employs additional analysis of semantic content and emotions expressed in reviews. In particular, we use the deep neural network to learn word embeddings representation and lexicon-based emotion indicators in order to integrate into the graph construction process. We demonstrate the effectiveness of BeGP and BeGPX on two real-world review datasets from Yelp.com. The results show that both approaches outperform state-of-the-art methods with accurately identifying fake review(er)s within the k-first order of rankings. In addition, BeGPX shows significant enhancement although being provided with only a few amount of learning labeled data.

Streaming graph partitioning methods have recently gained attention due to their ability to scale to very large graphs with limited resources. However, many such methods do not consider workload and graph characteristics. This may degrade the performance of queries by increasing inter-node communication and computational load imbalance. Moreover, existing workload-aware methods cannot consistently provide good performance as they do not consider dynamic workloads that keep emerging in graph applications. We address these issues by proposing a novel workload-adaptive streaming partitioner named WASP, that aims to achieve low-latency and high-throughput online graph queries. As each workload typically contains frequent query patterns, WASP exploits the existing workload to capture active vertices and edges which are frequently visited and traversed, respectively. This information is used to heuristically improve the quality of partitions either by avoiding the concentration of active vertices in a few partitions proportional to their visit frequencies or by reducing the probability of the cut of active edges proportional to their traversal frequencies. In order to assess the impact of WASP on a graph store and to show how easily the approach can be plugged on top of the system, we exploit it in a distributed graph-based RDF store. Our experiments over three synthetic and real-world graph datasets and the corresponding static and dynamic query workloads show that WASP achieves a better query performance against state-of-the-art graph partitioners, especially in dynamic query workloads.

The minimum cut problem, MC, and the special case of the vertex separator problem, consists in partitioning the set of nodes of a graph G into k subsets of given sizes in order to minimize the number of edges cut after removing the k-th set. Previous work on approximate solutions uses, in increasing strength and expense: eigenvalue, semidefinite programming, SDP, and doubly nonnegative, DNN, bounding techniques. In this paper, we derive strengthened SDP and DNN relaxations, and we propose a scalable algorithmic approach for efficiently evaluating, theoretically verifiable, both upper and lower bounds. Our stronger relaxations are based on a new gangster set, and we demonstrate how facial reduction, FR, fits in well to allow for regularized relaxations. Moreover, the FR appears to be perfectly well suited for a natural splitting of variables, and thus for the application of splitting methods. Here, we adopt the strictly contractive Peaceman-Rachford splitting method, sPRSM. Further, we bring useful redundant constraints back into the subproblems, and show empirically that this accelerates sPRSM.In addition, we employ new strategies for obtaining lower bounds and upper bounds of the optimal value of MC from approximate iterates of the sPRSM thus aiding in early termination of the algorithm. We compare our approach with others in the literature on random datasets and vertex separator problems. This illustrates the efficiency and robustness of our proposed method.

People of all ages enjoy solving geometric puzzles. However, finding suitable puzzles, e.g., puzzles with a moderate level of difficulty or puzzles with intellectually stimulating shapes can be difficult. In addition, designing innovative and appealing puzzles requires demanding effort and, typically, involves many trial and error processes. In this paper, we introduce a computational approach for designing geometric puzzles. Existing approaches employ bottom-up, constructive algorithms to generate puzzle pieces; therefore, intervening in the piece generation procedure is difficult. Differing from existing approaches that generate puzzles automatically or semi-automatically, we propose a top-down, partitioning-based approach, that enables us to control and edit piece shapes. With a subtle modification, the proposed algorithm can be easily extended to both 3D polycube and 2D polyomino puzzle design. To generate a variety of piece shapes, the proposed approach involves a capacity-constrained graph partitioning algorithm combined with polyomino tiling. We demonstrate the versatility of the proposed approach through various example designs, including fabricated puzzles, created using the proposed method.

Since p-spectral clustering has good performance in many practical problems, it has attracted great attention. The Cheeger cut criterion is used in p-spectral clustering to do graph partition. However, due to the improper affinity measure and outliers, the original p-spectral clustering algorithm is not effective in dealing with manifold data. To solve this problem, we propose a manifold p-spectral clustering (M-pSC) using path-based affinity measure. First, we design a path-based affinity function to describe the complex structures of manifold data. This affinity function obeys the clustering assumption that the data pairs within the manifold structure share high affinities, and the data pairs between different manifold structures share low affinities. This will help us construct a good affinity matrix, which carry more category information of the points. Then we propose a M-pSC algorithm using the path-based affinity function. In the Cheeger cut criterion, the p-Laplacian matrix are constructed based on the manifold affinity function, and the final clustering results are obtained by using the eigenvectors of graph p-Laplacian. At last, the proposed algorithm is tested on several public data sets and the experiments show that our algorithm is adaptive to different manifold data. Compared with other popular clustering algorithms, our algorithm has good clustering quality and robustness.

Directed graphs are widely used to model data flow and execution dependencies in streaming applications. This enables the utilization of graph partitioning algorithms for the problem of parallelizing execution on multiprocessor architectures under hardware resource constraints. However due to program memory restrictions in embedded multiprocessor systems, applications need to be divided into parts without cyclic dependencies. We found that this can be done by a subsequent second graph partitioning step with an additional acyclicity constraint. We have four main contributions. First, we show that this more constrained version of the graph partitioning problem is NP-complete and present linear time heuristics. We then integrate them into an existing multi-level graph partitioning framework to better handle large graphs. This achieves a 9% reduction of the edge cut compared to the previous single-level algorithm. Based on this, we engineer an evolutionary algorithm to further reduce the cut, achieving a 30% reduction on average compared to the state of the art. Finally, we integrate the partitioning heuristics into a graph compiler for an embedded multiprocessor architecture and show that this can reduce the amount of communication for a real-world imaging application and thereby accelerate it by an average of 11%. It is shown that the compiler can emit optimized code for vastly different hardware platforms using the heuristics. In addition, we demonstrate how a custom fitness function for the evolutionary algorithm can be used to optimize other objectives like load balancing if the communication volume is not predominantly important on a given hardware platform.

We give a distributed algorithm in the CONGEST model for property testing of planarity with one-sided error in general (unbounded-degree) graphs. Following Censor-Hillel et al. (Proceedings of the 30th International Symposium on Distributed Computing, pp. 43–56, 2016), who recently initiated the study of property testing in the distributed setting, our algorithm gives the following guarantee: For a graph $$G = (V,E)$$ and a distance parameter $$\epsilon $$ , if G is planar, then every node outputs accept, and if G is $$\epsilon $$ -far from being planar (i.e., more than $$\epsilon \cdot |E|$$ edges need to be removed in order to make G planar), then with probability $$1-1/\mathrm{poly}(n)$$ at least one node outputs reject. The algorithm runs in $$O(\log |V|\cdot \mathrm{poly}(1/\epsilon ))$$ rounds, and we show that this result is tight in terms of the dependence on |V|. Our algorithm combines several techniques of graph partitioning and local verification of planar embeddings. Furthermore, we show how a main subroutine in our algorithm can be applied to derive additional results for property testing of cycle-freeness and bipartiteness, as well as the construction of spanners, in minor-free (unweighted) graphs.

Dynamic Topic Modeling (DTM) is the ultimate solution for extracting topics from short texts generated in Online Social Networks (OSNs) like Twitter. It requires to be scalable and to be able to account for sparsity and dynamicity of short texts. Current solutions combine probabilistic mixture models like Dirichlet Multinomial or Pitman-Yor Process with approximate inference approaches like Gibbs Sampling and Stochastic Variational Inference to, respectively, account for dynamicity and scalability of DTM. However, these methods basically rely on weak probabilistic language models, which do not account for sparsity in short texts. In addition, their inference is based on iterative optimizations, which have scalability issues when it comes to DTM. We present GDTM, a single-pass graph-based DTM algorithm, to solve the problem. GDTM combines a context-rich and incremental feature representation method with graph partitioning to address scalability and dynamicity and uses a rich language model to account for sparsity. We run multiple experiments over a large-scale Twitter dataset to analyze the accuracy and scalability of GDTM and compare the results with four state-of-the-art models. In result, GDTM outperforms the best model by $$11\%$$ 11 % on accuracy and performs by an order of magnitude faster while creating four times better topic quality over standard evaluation metrics.

We consider an extended version of the classical Max-$$k$$ k -Cut problem in which we additionally require that the parts of the graph partition are connected. For this problem we study two alternative mixed-integer linear formulations and review existing as well as develop new branch-and-cut techniques like cuts, branching rules, propagation, primal heuristics, and symmetry breaking. The main focus of this paper is an extensive numerical study in which we analyze the impact of the different techniques for various test sets. It turns out that the techniques from the existing literature are not sufficient to solve an adequate fraction of the test sets. However, our novel techniques significantly outperform the existing ones both in terms of running times and the overall number of instances that can be solved.

The problem of dividing political territories in electoral process is a very important factor which contributes to the development of democracy in modern political systems. The most significant criteria for fairness of electoral process are demographic, geographic and political. Demographic criterion in the first place refers to the population equality, while the geographic one is mostly represented by compactness, contiguity and integrity. In this paper we propose a new integer linear programming formulation for the problem of political districting. The model is based on the graph representation of political territory, where territorial units are vertices and direct links between them are edges. The correctness of integer linear programming formulation is mathematically proven. In contrast to the most of the previous formulations, all three major criteria, population equality, compactness and contiguity, are completely taken into consideration. There are two models, one which deals with afore mentioned criteria where compactness is taken as an objective function, and the other one which takes into account interests of the decision maker, i.e. the political ruling body which organizes elections. Several numerical examples for the presented models are given which illustrate general aspects of the problem. The experimental results are obtained using CPLEX solver.

Clustering of single-cell RNA sequencing (scRNA-seq) data enables discovering cell subtypes, which is helpful for understanding and analyzing the processes of diseases. Determining the weight of edges is an essential component in graph-based clustering methods. While several graph-based clustering algorithms for scRNA-seq data have been proposed, they are generally based on k-nearest neighbor (KNN) and shared nearest neighbor (SNN) without considering the structure information of graph. Here, to improve the clustering accuracy, we present a novel method for single-cell clustering, called structural shared nearest neighbor-Louvain (SSNN-Louvain), which integrates the structure information of graph and module detection. In SSNN-Louvain, based on the distance between a node and its shared nearest neighbors, the weight of edge is defined by introducing the ratio of the number of the shared nearest neighbors to that of nearest neighbors, thus integrating structure information of the graph. Then, a modified Louvain community detection algorithm is proposed and applied to identify modules in the graph. Essentially, each community represents a subtype of cells. It is worth mentioning that our proposed method integrates the advantages of both SNN graph and community detection without the need for tuning any additional parameter other than the number of neighbors. To test the performance of SSNN-Louvain, we compare it to five existing methods on 16 real datasets, including nonnegative matrix factorization, single-cell interpretation via multi-kernel learning, SNN-Cliq, Seurat and PhenoGraph. The experimental results show that our approach achieves the best average performance in these datasets.

Mobile Cloud Computing (MCC) aims at leveraging remote resources to boost application performance on mobile devices while conserving resources such as battery, memory, and storage. Offloading computations and outsourcing tasks are, however, associated with numerous challenges known from distributed systems. Typical mobile applications have a monolithic design and are not laid out for a distributed deployment and execution. In this work, we present how to design and partition such applications and how these partitions stay synchronized in a cost-efficient manner at runtime. We introduce our comprehensive and extendable framework MOCCAA (MObile Cloud Computing AdaptAble) that supports developers along this path. Its performance gain is mainly achieved through a new graph partitioning heuristic that is searching for the maximally beneficial cut, through minimized monitoring efforts for resource consumption prediction, through scalable and location-aware resource discovery and management, and through our graph-based delta synchronization of local and remote object states. In particular, the graph partitioning heuristic and the delta synchronization allow us to reduce synchronization costs and improve quality dimensions such as latency and bandwidth consumption.

Graph coloring is a manifestation of graph partitioning, wherein a graph is partitioned based on the adjacency of its elements. The fact that there is no general efficient solution to this problem that may work unequivocally for all graphs opens up the realistic scope for combinatorial optimization algorithms to be invoked. The algorithmic complexity of graph coloring is non-deterministic in polynomial time and hard. To the best of our knowledge, there is no algorithm as yet that procures an exact solution of the chromatic number comprehensively for any and all graphs within the polynomial (P) time domain. Here, we present a novel heuristic, namely the ‘trailing path’, which returns an approximate solution of the chromatic number within P time, and with a better accuracy than most existing algorithms. The ‘trailing path’ algorithm is effectively a subtle combination of the search patterns of two existing heuristics (DSATUR and largest first) and operates along a trailing path of consecutively connected nodes (and thereby effectively maps to the problem of finding spanning tree(s) of the graph) during the entire course of coloring, where essentially lies both the novelty and the apt of the current approach. The study also suggests that the judicious implementation of randomness is one of the keys toward rendering an improved accuracy in such combinatorial optimization algorithms. Apart from the algorithmic attributes, essential properties of graph partitioning in random and different structured networks have also been surveyed, followed by a comparative study. The study reveals the remarkable stability and absorptive property of chromatic number across a wide array of graphs. Finally, a case study is presented to demonstrate the potential use of graph coloring in protein design—yet another hard problem in structural and evolutionary biology.

The minimum k-partition problem is a challenging combinatorial problem with a diverse set of applications ranging from telecommunications to sports scheduling. It generalizes the max-cut problem and has been extensively studied since the late sixties. Strong integer formulations proposed in the literature suffer from a large number of constraints and variables. In this work, we introduce two more compact integer linear and semidefinite reformulations that exploit the sparsity of the underlying graph and develop theoretical results leveraging the power of chordal decomposition. Numerical experiments show that the new formulations improve upon state-of-the-art.

Differential privacy protection model provides strict and quantitative risk representation for privacy disclosure, which greatly ensures the availability of data. However, most existing methods do not consider the semantic context, so they are vulnerable to attacks based on semantic information. Therefore, dynamic social privacy protection based on graph pattern partitioning is designed to satisfy differential privacy protection. Firstly, the structure of social network is represented as a graph model, and the original graph is classified into several sub-graphs according to the characteristics of nodes. Then, the dense area of each sub-graph is divided by quad-tree method, and the noise of differential privacy protection is added to the leaf nodes of the tree, and the graph publishing is generated by sub-graph reconstruction. Finally, the feasibility and practicability of the model are verified by statistical analysis, such as degree distribution, shortest path, and clustering coefficient. The simulation results show the validity and applicability of the privacy protection method proposed in this paper.

The increasing utility of specialized circuits and growing applications of optimization call for the development of efficient hardware accelerator for solving optimization problems. Hopfield neural network is a promising approach for solving combinatorial optimization problems due to the recent demonstrations of efficient mixed-signal implementation based on emerging non-volatile memory devices. Such mixed-signal accelerators also enable very efficient implementation of various annealing techniques, which are essential for finding optimal solutions. Here we propose a "weight annealing" approach, whose main idea is to ease convergence to the global minima by keeping the network close to its ground state. This is achieved by initially setting all synaptic weights to zero, thus ensuring a quick transition of the Hopfield network to its trivial global minima state and then gradually introducing weights during the annealing process. The extensive numerical simulations show that our approach leads to a better, on average, solutions for several representative combinatorial problems compared to prior Hopfield neural network solvers with chaotic or stochastic annealing. As a proof of concept, a 13-node graph partitioning problem and a 7-node maximum-weight independent set problem are solved experimentally using mixed-signal circuits based on, correspondingly, a 20 x 20 analog-grade TiO2 memristive crossbar and a 12 x 10 eFlash memory array. Funding Agencies|Semiconductor Research Corporation (SRC) funded JUMP CRISP center, NSF/SRC E2CDA grant [1740352]; DENSO CORPORATION

Graph partition is a classical combinatorial optimization and graph theory problem, and it has a lot of applications, such as scientific computing, VLSI design and clustering etc. In this paper, we study the partition problem on large scale directed graphs under a new objective function, a new instance of graph partition problem. We firstly propose the modeling of this problem, then design an algorithm based on multi-level strategy and recursive partition method, and finally do a lot of simulation experiments. The experimental results verify the stability of our algorithm and show that our algorithm has the same good performance as METIS. In addition, our algorithm is better than METIS on unbalanced ratio.

The emergence of collaborative interactions has empowered users by enabling their interactions through tagging practices that create a folksonomy, also called, classification of the shared resources, any identifiable thing or item on the system. In education, tagging is considered a powerful meta-cognitive strategy that successfully engages learners in the learning process. Besides, the collaborative tagging gathers learners’ opinions, thus, provides more comprehensible recommendations. Still, the abundant shared contents are mostly unorganized which makes it hard for users to select and discover the appropriate items of their interests. Thus, the use of recommender systems overcomes the distressing search problem by assisting users in their searching and exploring experience, and suggesting relevant items matching their preferences. In this regard, this article presents a folksonomy graphs based context-aware recommender system (CARS) of annotated books. The generated graphs express the semantic relatedness between these resources, i.e. books, by effectively modeling the folksonomy relationship between user-resource-tag and integrating contextual information within a multi-layer graph referring to a Knowledge Graph (KG). To put our proposal into shape, we model a real-world application of Goodbooks-10k dataset to recommend books. The proposed approach incorporates spectral clustering to deal with the graph partitioning problem. The experimental evaluation shows relevant performance results of graph-based book recommendations.

Graph convolutional networks (GCNs) have been applied successfully in social networks and recommendation systems to analyze graph data. Unlike conventional neural networks, GCNs introduce an aggregation phase, which is both computation- and memory-intensive. This phase aggregates features from the neighboring vertices in the graph, which incurs significant amounts of irregular data and memory access. The emerging computation-in-memory (CIM) architecture presents a promising solution to alleviate the problem of irregular accesses and provide fast near-data processing for GCN applications by integrating both three-dimensional stacked CIM and general-purpose processing units in the system. This paper presents Graph-CIM, which exploits the hybrid CIM architecture to determine the allocation of GCN applications. Graph-CIM models the GCN application process as a directed acyclic graph (DAG) and allocates tasks on the hybrid CIM architecture. It achieves fine-grained graph partitioning to capture the irregular characteristics of the aggregation phase of GCN applications. We use a set of representative GCN models and standard graph datasets to evaluate the effectiveness of Graph-CIM. The experimental results show that Graph-CIM can significantly reduce the processing latency and data-movement overhead compared with the representative schemes.

Graph coloring has a wide range of real world applications, such as in the operations research, communication network, computational biology and compiler optimization fields. In our recent work [1], we propose a divide-and-conquer approach for graph coloring, called VColor. Such an approach has three generic subroutines. (i) Graph partition subroutine: VColor partitions a graph G into a vertex cut partition (VP), which comprises a vertex cut component (VCC) and small non-overlapping connected components (CCs). (ii) Component coloring subroutine: VColor colors the VCC and the CCs by efficient algorithms. (iii) Color combination subroutine: VColor combines the local colors by exploiting the maximum matchings of color combination bigraphs (CCBs). VColor has revealed some major bottlenecks of efficiency in these subroutines. Therefore, in this paper, we propose VColor*, an approach which addresses these efficiency bottlenecks without using more colors both theoretically and experimentally. The technical novelties of this paper are the following. (i) We propose the augmented VP to index the crossing edges of the VCC and the CCs and propose an optimized CCB construction algorithm. (ii) For sparse CCs, we propose using a greedy coloring algorithm that is of polynomial time complexity in the worst case, while preserving the approximation ratio. (iii) We propose a distributed graph coloring algorithm. Our extensive experimental evaluation on real-world graphs confirms the efficiency of VColor*. In particular, VColor* is 20X and 50X faster than VColor and uses the same number of colors with VColor on the Pokec and PA datasets, respectively. VColor* also significantly outperforms the state-of-the-art graph coloring methods.

Clustering ensemble has been increasingly popular in the recent years by consolidating several base clustering methods into a probably better and more robust one. However, cluster dependability has been ignored in the majority of the presented clustering ensemble methods that exposes them to the risk of the low-quality base clustering methods (and consequently the low-quality base clusters). In spite of some attempts made to evaluate the clustering methods, it seems that they consider each base clustering individually regardless of the diversity. In this study, a new clustering ensemble approach has been proposed using a weighting strategy. The paper has presented a method for performing consensus clustering by exploiting the cluster uncertainty concept. Indeed, each cluster has a contribution weight computed based on its undependability. All of the predicted cluster tags available in the ensemble are used to evaluate a cluster undependability based on an information theoretic measure. The paper has proposed two measures based on cluster undependability or uncertainty to estimate the cluster dependability or certainty. The multiple clusters are reconciled through the cluster uncertainty. A clustering ensemble paradigm has been proposed through the proposed method. The paper has proposed two approaches to achieve this goal: a cluster-wise weighted evidence accumulation and a cluster-wise weighted graph partitioning. The former approach is based on hierarchical agglomerative clustering and co-association matrices, while the latter is based on bi-partite graph formulating and partitioning. In the first step of the former, the cluster-wise weighing co-association matrix is proposed for representing a clustering ensemble. The proposed approaches have been then evaluated on 19 real-life datasets. The experimental evaluation has revealed that the proposed methods have better performances than the competing methods; i.e. through the extensive experiments on the real-world datasets, it has been concluded that the proposed method outperforms the state-of-the-art. The substantial experiments on some benchmark data sets indicate that the proposed methods can effectively capture the implicit relationship among the objects with higher clustering accuracy, stability, and robustness compared to a large number of the state-of-the-art techniques, supported by statistical analysis.

In short-range molecular dynamics (MD) simulations, inhomogeneous particle distributions that dynamically change over time require flexible load-balancing methods to achieve good parallel efficiency. We have realized a general framework that can support different load-balancing methods and that can extend existing simulation packages in a minimally invasive way. This is a follow-up to recent work where we integrated it into the MD software ESPResSo to support load-balancing. We have realized a first partitioning strategy based on space-filling curves that can be used for efficient load-balanced multi-physics simulations. In this work we present a new graph-based partitioning strategy that leads to unstructured spatial domain decompositions and integrates well into the existing framework. We apply this to an inhomogeneous soot agglomeration scenario. For several load metrics, graph partitioning leads to better results than space-filling curves. The results indicate that the parallel performance for a given scenario requires a delicate combination of partitioning strategy and load metrics.

This paper presents a new mathematical programming model and a solution approach for a special class of graph partitioning problem. The problem studied here is in the context of distributed web search, in which a very large world-wide-web graph is partitioned to improve the efficiency of webpage ranking (known as PageRank). Although graph partitioning problems have been widely studied and there have been several computational algorithms and mathematical programming models in the literature, the graph partitioning problem for PageRank imposes unique constraints on the density balance. This problem is called the min-cut density-balanced partitioning problem. In this paper, we propose a new mathematical programming model and a solution approach to efficiently solve this min-cut density-balanced partitioning problem. As the objective on the minimum cut and the constraint on the density balance are not the direct performance measure of PageRank, we also investigate the performance of the solutions obtained from a MIP solver and our approach on the ranking’s accuracy and the local ranking’s computation times. The experiment results show both solutions are comparable in terms of the ranking’s accuracy and the local ranking’s computation times whereas it is much faster to obtain the partitioning solutions using our approach.

Relations among data entities in most big data sets can be modeled by a big graph. Implementation and execution of algorithms related to the structure of big graphs is very important in different fields. Because of the inherently high volume of big graphs, their calculations should be performed in a distributed manner. Some distributed systems based on vertex-centric model have been introduced for big graph calculations in recent years. The performance of these systems in terms of run time depends on the partitioning and distribution of the graph. Therefore, the graph partitioning is a major concern in this field. This paper concentrates on big graph partitioning approaches for distribution of graphs in vertex-centric systems. This briefly discusses vertex-centric systems and formulates different models of graph partitioning problem. Then, a review of recent methods of big graph partitioning for these systems is shown. Most recent methods of big graph partitioning for vertex centric systems can be categorized into three classes: (i) stream-based methods that see vertices or edges of the graph in a stream and partition them, (ii) distributed methods that partition vertices or edges in a distributed manner, and (iii) dynamic methods that change partitions during the execution of algorithms to obtain better performance. This study compares the properties of different approaches in each class and briefly reviews methods that are not in these categories. This comparison indicates that The streaming methods are good choices for initial load of the graph in Vertex-centric systems. The distributed and dynamic methods are appropriate for long-running applications.

Similarity join on time series databases is an essential operation for data analysis applications. Due to the curse of dimensionality, it is not suitable to use traditional index techniques, such as R-tree and kd-tree. In the paper, a dynamic segment index (i.e., DSTree) is utilized to reduce the huge comparison cost on the similarity join on time series databases. However, the DSTree is designed for similarity search and only supports bound estimations between a time series and a batch of time series in a DSTree node. To make the DSTree suitable for the similarity join on time series databases, it is necessary to have tight bounds for the nodes to achieve a better pruning power, where the biggest challenge is that the DSTree nodes may have different segmentations. To solve the problem aforementioned, a segmentation alignment and synopsis evaluation method is proposed to support the estimation of DSTree nodes to significantly reduce the time cost by pruning unnecessary comparisons. Moreover, to make our approach I/O efficient, a caching strategy is proposed by taking advantage of both graph partitioning and the locality of the DSTree index. The efficiency and effectiveness of the proposed approaches are verified by experiments on real-life datasets.

Relations among data items can be modeled with graphs in most of big data sets such as social networks’ data. This modeling creates big graphs with many vertices and edges. Balanced k-way graph partitioning is a common problem with big graphs. It has many applications in several fields. There are many approximate solutions for this problem; however, most of them do not have enough scalability for big graph partitioning and cannot be executed in a distributed manner. Vertex-centric model has been introduced recently as a scalable distributed processing method for big graphs. There are a few methods for graph partitioning based on this model. Existing approaches only consider one-step neighbors of vertices for graph partitioning and do not consider neighbors with higher steps. In this paper, a distributed method is introduced based on vertex-centric model for balanced k-way graph partitioning. This method applies the personalized PageRank vectors of vertices and partitions to decide how vertices are joined partitions. This method has been implemented in the Giraph system. The proposed method has been evaluated with several synthetic and real graphs. Experimental results have shown that this method has scalability for partitioning big graphs. It was also found that this method produces partitions with higher quality compared to the state-of-the-art stream-based methods and distributed methods based on vertex-centric programming model. Its result is close to the results of Metis method.

Social networks offer various services such as recommendations of social events, or delivery of targeted advertising material to certain users. In this work, we focus on a specific type of services modeled as constrained graph partitioning (CGP). CGP assigns users of a social network to a set of classes with bounded capacities so that the similarity and the social costs are minimized. The similarity cost is proportional to the dissimilarity between a user and his class, whereas the social cost is measured in terms of friends that are assigned to different classes. In this work, we investigate two solutions for CGP. The first utilizes a game-theoretic framework, where each user constitutes a player that wishes to minimize his own social and similarity cost. The second employs local search, and aims at minimizing the global cost. We show that the two approaches can be unified under a common agent-based framework that allows for two types of deviations. In a unilateral deviation, an agent switches to a new class, whereas in a bilateral deviation a pair of agents exchange their classes. We develop a number of optimization techniques to improve result quality and facilitate efficiency. Our experimental evaluation on real datasets demonstrates that the proposed methods always outperform the state of the art in terms of solution quality, while they are up to an order of magnitude faster.

Graph partitioning methods in data mining have been widely used to discover protein complexes in protein–protein interaction (PPI) network. However, PPI networks with attributes need more effective attribute graph partitioning methods. Attribute graph partitioning aims to obtain high quality partitions satisfying the requirement: nodes in the same partition not only connect to each other more densely but also share more similar attribute values. In this paper, we propose a novel method for attributed graph partitioning based on fuzzy clustering. This method firstly devises a unified similarity measure using SimRank to construct the fuzzy similarity matrix of the attributed graph and can integrate structural and attribute similarities of nodes into a flexible weighted framework. Then it deduces the corresponding fuzzy equivalent matrix using fuzzy set theory. Finally, the result of partitioning can be obtained using fuzzy clustering algorithm. We conduct some experiments on several typical attributed graphs, which can also simulate PPI networks with attributes. The results show that our method is very effective to identify high quality partitions of attributed graphs and even performs better than some representative methods.

As the underlying database schemas become larger and more complex, it is difficult for casual users to understand the schemas and contents of databases. Therefore, it has become an essential task to summarize the database schemas. However, most prior approaches pay little attention to the topological characteristics between tables, ignore the effect of the user feedback, and fail to accurately predict the number of clusters in the output. This seriously limits their accuracy of schema summarization. To deal with the problems, we propose a new schema summarization method based on a graph partition mechanism. First, we introduce a novel strategy to construct a similarity matrix between tables, which is based on the topology compactness, content similarity and query logs. Then we provide a calculation formula for table importance and a detection scheme of the most important nodes in local areas. Both are used for selecting the initial cluster centers and predicting the number of clusters in the graph partition mechanism. Finally, we evaluate the proposed method over the database TPC-E, and results demonstrate that it achieves high performance in summarizing accuracy.

The bulk synchronous parallel (BSP) model is very user friendly for coding and debugging parallel graph algorithms. However, existing BSP-based distributed graph-processing frameworks, such as Pregel, GPS and Giraph, routinely suffer from high communication costs. These high communication costs mainly stem from the fine-grained message-passing communication model. In order to address this problem, we propose a new computation model with low communication costs, called LCC-BSP. We use this model to design and implement a high-performance distributed graph-processing framework called LCC-Graph. This framework eliminates high communication costs in existing distributed graph-processing frameworks. Moreover, LCC-Graph also balances the computation workloads among all compute nodes by optimizing graph partitioning, significantly reducing the computation time for each superstep. Evaluation of LCC-Graph on a 32-node cluster, driven by real-world graph datasets, shows that it significantly outperforms existing distributed graph-processing frameworks in terms of runtime, particularly when the system is supported by a high-bandwidth network. For example, LCC-Graph achieves an order of magnitude performance improvement over GPS and GraphLab.

In this paper we present a novel moment-based skeleton detection for representing human objects in RGB-D videos with animated 3D skeletons. An object often consists of several parts, where each of them can be concisely represented with a skeleton. However, it remains as a challenge to detect the skeletons of individual objects in an image since it requires an effective part detector and a part merging algorithm to group parts into objects. In this paper, we present a novel fully unsupervised learning framework to detect the skeletons of human objects in a RGB-D video. The skeleton modeling algorithm uses a pipeline architecture which consists of a series of cascaded operations, i.e., symmetry patch detection, linear time search of symmetry patch pairs, part and symmetry detection, symmetry graph partitioning, and object segmentation. The properties of geometric moment-based functions for embedding symmetry features into centers of symmetry patches are also investigated in detail. As compared with the state-of-the-art deep learning approaches for skeleton detection, the proposed approach does not require tedious human labeling work on training images to locate the skeleton pixels and their associated scale information. Although our algorithm can detect parts and objects simultaneously, a pre-learned convolution neural network (CNN) can be used to locate the human object from each frame of the input video RGB-D video in order to achieve the goal of constructing real-time applications. This much reduces the complexity to detect the skeleton structure of individual human objects with our proposed method. Using the segmented human object skeleton model, a video surveillance application is constructed to verify the effectiveness of the approach. Experimental results show that the proposed method gives good performance in terms of detection and recognition using publicly available datasets.

Noise is irrelevant or meaningless data and hinders most types of data analysis. The existing clustering algorithms seldom take the noise points into consideration and cannot detect arbitrary-shaped clusters. This paper presents a Hierarchical Clustering algorithm Based on Noise Removal (HCBNR). It is robust against noise points and good at discovering clusters with arbitrary shapes. In this work, natural neighbor-based density is applied to remove noise points in a data set firstly. Then we construct a saturated neighbor graph on the rest points, and a novel modularity-based graph partitioning algorithm is used to divide the graph into small clusters. Finally, the small clusters are repeatedly merged according to a novel similarity metric between clusters until the desired cluster number is obtained. The experimental results on synthetic data sets and real data sets show that our method can accurately identify noise points and obtain better clustering results than existing clustering algorithms when discovering arbitrary-shaped clusters.

It has been shown that the adjacency eigenspace of a network contains key information of its underlying structure. However, there has been no study on spectral analysis of the adjacency matrices of directed signed graphs. In this paper, we derive theoretical approximations of spectral projections from such directed signed networks using matrix perturbation theory. We use the derived theoretical results to study the influences of negative intra cluster and inter cluster directed edges on node spectral projections. We then develop a spectral clustering based graph partition algorithm, SC-DSG, and conduct evaluations on both synthetic and real datasets. Both theoretical analysis and empirical evaluation demonstrate the effectiveness of the proposed algorithm.

Community discovery is an essential topic in social network analysis since it provides a way for recursively decomposing a large social graph to easily interpretable subgraphs. The implementation of four major community discovery algorithms, namely the Newman–Girvan or Edge Betweeness, the Walktrap, the Louvain, and the CNM as Java analytics over Neo4j is described. Their correctness was evaluated functionally in two real Twitter graphs with vastly different characteristics. This was done on the grounds that a successful structural graph partitioning should eventually be reflected in the network functionality domain. Additionally, most real world graphs lack a list of ground truth communities, rendering a structural verification difficult, while functionality can be easily observed in most cases. Naturally, this renders the evaluation network-specific, as different social networks have different operational characteristics. The primary algorithmic finding was that the Louvain algorithm yields Twitter communities whose distribution size matches closer, in terms of the Kullback–Leibler divergence, the tweet and retweet distributions, with Newman–Girvan, Walktrap, and CNM following in that order.

Character segmentation is an important task in optical character recognition (OCR). The quality of any OCR system is highly dependent on character segmentation algorithm. Despite the availability of various character segmentation methods proposed to date, existing methods cannot satisfyingly segment characters belonging to some complex writing styles such as the Lanna Dhamma characters. In this paper, a new character segmentation method named graph partitioning-based character segmentation is proposed to address the problem. The proposed method can deal with multi-level writing style as well as touching and broken characters. It is considered as a generalization of existing approaches to multi-level writing style. The proposed method consists of three phases. In the first phase, a newly devised over-segmentation technique based on morphological skeleton is used to obtain redundant fragments of a word image. The fragments are then used to form a segmentation hypotheses graph. In the last phase, the hypotheses graph is partitioned into subgraphs each corresponding to a segmented character using the partitioning algorithm developed specifically for character segmentation purpose. Experimental results based on handwritten Lanna Dhamma characters datasets showed that the proposed method achieved high correct segmentation rate and outperformed existing methods for the Lanna Dhamma alphabet.

In order to process complex and large-scale graph data, numerous distributed graph-parallel computing platforms have been proposed. PowerGraph is an excellent representative of them. It has exhibited better performance, such as faster graph-processing rate and higher scalability, than others. However, like in other distributed graph computing systems, unnecessary and excessive communications among computing nodes in PowerGraph not only aggravate the network I/O workload of the underlying computing hardware systems but may also cause a decrease in runtime performance. In this paper, we propose and implement a mechanism called L-PowerGraph, which reduces the communication overhead in PowerGraph. First, L-PowerGraph identifies and eliminates the avoidable communications in PowerGraph. Second, in order to further reduce the required communications L-PowerGraph proposes an edge direction-aware master appointment strategy, in which L-PowerGraph appoints the replica with both incoming and outgoing edges as master. Third, L-PowerGraph proposes an edge direction-aware graph partition strategy, which optimally isolates the outgoing edges from the incoming edges of a vertex during the graph partition process. We have conducted extensive experiments using real-world datasets, and our results verified the effectiveness of the proposed mechanism. For example, compared with PowerGraph under Random partition scenario L-PowerGraph can not only reduce up to 30.5% of the communication overhead but also cut up to 20.3% of the runtime for PageRank algorithm while processing Live-journal dataset. The performance improvement achieved by L-PowerGraph over our precursor work, LightGraph, which only reduces the synchronizing communication overhead, is also verified by our experimental results.

The publication of this article unfortunately contained a mistake. The assignment to the affiliations of author Satish Karra was not correct

We study packing problems with matroid structures, which includes the strength of a graph of Cunningham and scheduling problems. If $$\mathcal {M}$$ is a matroid over a set of elements S with independent set $$\mathcal {I}$$ , and $$m=|S|$$ , we suppose that we are given an oracle function that takes an independent set $$A\in \mathcal {I}$$ and an element $$e\in S$$ and determines if $$A\cup \{e\}$$ is independent in time I(m). Also, given that the elements of A are represented in an ordered way $$A=\{A_1,\dots ,A_k\}$$ , we denote the time to check if $$A\cup \{e\}\notin \mathcal {I}$$ and if so, to find the minimum $$i\in \{0,\dots ,k\}$$ such that $$\{A_1,\dots ,A_i\}\cup \{e\}\notin \mathcal {I}$$ by $$I^*(m)$$ . Then, we describe a new FPTAS that computes for any $$\varepsilon >0$$ and for any matroid $$\mathcal {M}$$ of rank r over a set S of m elements, in memory space O(m), the packing $$\varLambda ({\mathcal {M}})$$ within $$1+\varepsilon $$ in time $$O(mI^*(m)\log (m)\log (m/r)/\varepsilon ^2)$$ , and the covering $$\varUpsilon ({\mathcal {M}})$$ in time $$O(r\varUpsilon ({\mathcal {M}})I(m)\log (m)\log (m/r)/\varepsilon ^2)$$ . This method outperforms in time complexity by a factor of $$\varOmega (m/r)$$ the FPTAS of Plotkin, Shmoys, and Tardos, and a factor of $$\varOmega (m)$$ the FPTAS of Garg and Konemann. On top of the value of the packing and the covering, our algorithm exhibits a combinatorial object that proves the approximation. The applications of this result include graph partitioning, minimum cuts, VLSI computing, job scheduling and others.

This paper presents a novel distributed object segmentation framework that allows one to extract potentially large coherent objects from digital images. The proposed approach requires minimum user supervision and permits to segment the objects accurately. It works in three steps starting with the user input in form of few mouse clicks on the target object. First, based on user input, the statistical characteristics of the target distributed object are modeled with Gaussian mixture model. This model serves as the primary segmentation of the object. In the second step, the segmentation result is refined by performing connected component analysis to reduce false positives. In the final step the resulting segmentation map is dilated to select the neighboring pixels that are potentially incorrectly classified; this allows us to recast the segmentation as a graph partitioning problem that can be solved using the well-known graph cut technique. Extensive experiments have been carried out on heterogeneous images to test the accuracy of the proposed method for the segmentation of various types of distributed objects. Examples of application of proposed technique in remote sensing to segment roads and rivers from aerial images are also presented. The visual and objective evaluation and comparison with the existing techniques show that the proposed tool can deliver optimal performance when applied to tough object segmentation tasks.

Unsupervised ensemble, or consensus clustering, consists in finding the optimal combination strategy of individual clusterings that is robust with respect to the selection of an algorithmic clustering pool. Recently an approach was proposed based on the concept of consensus graph that has profound advantages over its predecessors. Despite its robust properties this approach assigns the same weight to the contribution of each clustering to the final solution. In this paper, we propose a weighting policy for this problem that is based on internal clustering quality measures and compare against other popular approaches. Results on publicly available datasets show that weights can significantly improve the accuracy performance while retaining the robust properties.

In this paper we consider a problem of graph \({\mathcal P}\)-coloring consisting in partitioning the vertex set of a graph such that each of the resulting sets induces a graph in a given additive, hereditary class of graphs \({\mathcal P}\). We focus on partitions generated by the greedy algorithm. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs a \({\mathcal P}\)-coloring with a least k colors is \(\mathbb {NP}\)-complete for an infinite number of classes \({\mathcal P}\). On the other hand we get a polynomial-time certifying algorithm if k is fixed and the family of minimal forbidden graphs defining the class \({\mathcal P}\) is finite. We also prove \(\mathrm{co}\mathbb {NP}\)-completeness of the problem of deciding whether for a given graph G the difference between the largest number of colors used by the greedy algorithm and the minimum number of colors required in any \({\mathcal P}\)-coloring of G is bounded by a given constant. A new Brooks-type bound on the largest number of colors used by the greedy \({\mathcal P}\)-coloring algorithm is given.

In this paper, we continue investigating the partition dimension for disconnected graphs. We determine the partition dimension for some classes of disconnected graphs G consisting of two components. If $$G=G_1 \cup G_2$$ , then we give the bounds of the partition dimension of G for $$G_1 = P_n$$ or $$G_1=C_n$$ and also for $$pd(G_1)=pd(G_2)$$ .

Graphs are widely used to model complex data in many applications, such as bioinformatics, chemistry, social networks, pattern recognition. A fundamental and critical query primitive is to efficiently search similar structures in a large collection of graphs. This article mainly studies threshold-based graph similarity search with edit distance constraints. Existing solutions to the problem utilize fixed-size overlapping substructures to generate candidates, and thus become susceptible to large vertex degrees and distance thresholds. In this article, we present a partition-based approach to tackle the problem. By dividing data graphs into variable-size non-overlapping partitions, the edit distance constraint is converted to a graph containment constraint for candidate generation. We develop efficient query processing algorithms based on the novel paradigm. Moreover, candidate-pruning techniques and an improved graph edit distance verification algorithm are developed to boost the performance. In addition, a cost-aware graph partitioning method is devised to optimize the index. Extending the partition-based filtering paradigm, we present a solution to the top- $$k$$ graph similarity search problem, where tailored filtering, look-ahead and computation-sharing strategies are exploited. Using both public real-life and synthetic datasets, extensive experiments demonstrate that our approaches significantly outperform the baseline and its alternatives.

In the past several years, lots of out-of-core graph processing systems are built to process big graph datasets in computer systems with limited main memory. Due to the iterative nature of graph algorithms, most of these systems employ synchronous execution model to organize the computation, i.e., divide the computing into multiple rounds, each of which corresponds to one iteration of the graph algorithm. In order to fully utilize the disk bandwidth, these systems sequentially scan the whole graph dataset at each iteration. However, as the graph dataset under processing may be huge, more iterations generally means larger I/O overheads. Although asynchronous implementation of the synchronous execution model allows message passing within an iteration, the effectiveness is still limited. Since in such model, at most one message is allowed to be passed from one vertex to another. In this paper, we investigate the idea of block updating in the synchronous execution model framework in the out-of-core graph processing systems. With this new model, the system conducts graph algorithm on the loaded subgraph (i.e., block) to its local convergence, and then switches to other subgraphs to continue this process, until global convergence is reached. We implement this new model in GraphChi (the result system is called BlockGraphChi), and propose a companion graph partition method, named as DMLP. By this study, we found that compared with the original execution model of GraphChi: (1) the new model can generally reduce the amount of iterations (and thus the I/O overheads) for graph algorithms, while the extent of reduction depends on the method of graph partitioning and the properties of the algorithms; (2) the new model can dramatically reduce the overall execution time of graph traversal algorithms (by up to 31.4 $$\times $$ ), and better partitioning method leads to better performance; (3) the new model has much smaller effectiveness on improving the overall performance of fix-point algorithms, such as PageRank, due to the increased computational overhead.

The Vertex Separator Problem for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets that satisfy specified size constraints. The Vertex Separator Problem was formulated in the paper 10.1016/j.ejor.2014.05.042 as a continuous (non-concave/non-convex) bilinear quadratic program. In this paper, we develop a more general continuous bilinear program which incorporates vertex weights, and which applies to the coarse graphs that are generated in a multilevel compression of the original Vertex Separator Problem. We develop a method for improving upon a given vertex separator by applying a Mountain Climbing Algorithm to the bilinear program using an incidence vector for the separator as a starting guess. Sufficient conditions are developed under which the algorithm can improve upon the starting guess after at most two iterations. The refinement algorithm is augmented with a perturbation technique to enable escapes from local optima and is embedded in a multilevel framework for solving large scale instances of the problem. The multilevel algorithm is shown through computational experiments to perform particularly well on communication and collaboration networks.

Graph clustering (partitioning) is a helpful tool in understanding complex systems and analyzing their structure and internal properties. One approach for graph clustering is based on partitioning the graph into cliques. However, clique models are too restrictive and prone to errors given imperfect data. Thus, using clique relaxations instead may provide a more reasonable and applicable partitioning of the graph. An s-club is a distance-based relaxation of a clique and is formally defined as a subset of vertices inducing a subgraph with a diameter of at most s. In this work, we study the minimum s-club partitioning problem, which is to partition the graph into a minimum number of non-overlapping s-club clusters. Integer programming techniques and combinatorial branch-and-bound framework are employed to develop exact algorithms to solve this problem. We also study and compare the computational performance of the proposed algorithms for the special cases of $$s=2$$ and $$s=3$$ on a test-bed of randomly generated instances and real-life graphs.