Graph Partitioning Research Articles

Polynomial bounds for chromatic number VIII. Excluding a path and a complete multipartite graph

AbstractWe prove that for every path , and every integer , there is a polynomial such that every graph with chromatic number greater than either contains as an induced subgraph, or contains as a subgraph the complete ‐partite graph with parts of cardinality . For and general this is a classical theorem of Gyárfás, and for and general this is a theorem of Bonamy et al.

Load Balanced PIM-Based Graph Processing

Graph processing is widely used for many modern applications, such as social networks, recommendation systems, and knowledge graphs. However, processing large-scale graphs on traditional Von Neumann architectures is challenging due to the irregular graph data and memory-bound graph algorithms. Processing-in-memory (PIM) architecture has emerged as a promising approach for accelerating graph processing by enabling computation to be performed directly on memory. Despite having many processing units and high local memory bandwidth, PIM often suffers from insufficient global communication bandwidth and high synchronization overhead due to load imbalance. This article proposes GraphB, a novel PIM-based graph processing system, to address all these issues. From the algorithm perspective, we propose a degree-aware graph partitioning algorithm that can generate balanced partitioning at a low cost. From the architecture perspective, we introduce tile buffers incorporated with an on-chip 2D-Mesh, which provides high bandwidth for inter-node data transfer. Dataflow in GraphB is designed to enable computation–communication overlap and dynamic load balancing. In a PyMTL3-based cycle-accurate simulator with five real-world graphs and three common algorithms, GraphB achieves an average 2.2× and maximum 2.8× speedup compared to the SOTA PIM-based graph processing system GraphQ.

Reordering Functions in Mobiles Apps for Reduced Size and Faster Start-Up

Function layout, also known as function reordering or function placement, is one of the most effective profile-guided compiler optimizations. By reordering functions in a binary, compilers can improve the performance of large-scale applications or reduce the compressed size of mobile applications. Although the technique has been extensively studied in the context of large-scale binaries, no study has thoroughly investigated function layout algorithms on mobile applications. In this article, we develop the first principled solution for optimizing function layouts in the mobile space. To this end, we identify two key optimization goals: reducing the compressed code size and improving the cold start-up time of a mobile application. Then, we propose a formal model for the layout problem, whose objective closely matches our goals, and a novel algorithm for optimizing the layout. The method is inspired by the classic balanced graph partitioning problem. We have carefully engineered and implemented the algorithm in an open-source compiler, Low-level Virtual Machine (LLVM). An extensive evaluation of the new method on large commercial mobile applications demonstrates improvements in start-up time and compressed size compared to the state-of-the-art approach. 1

PECC: parallel expansion based on clustering coefficient for efficient graph partitioning

PECC: parallel expansion based on clustering coefficient for efficient graph partitioning

Optimal multi-objective sizing of renewable energy sources and battery energy storage systems for formation of a multi-microgrid system considering diverse load patterns

Optimal microgrid design is pivotal in planning active distribution networks (ADNs) with intermittent renewable energy sources (RESs) and battery energy storage systems (BESSs). This paper introduces an innovative approach to clustering existing ADN systems, incorporating RESs and BESSs into a set of microgrids (MGs) termed a multi-microgrid (MMG). The approach accommodates diverse load patterns, considering the dynamic nature of residential, commercial, and industrial loads. The proposed methodology employs an innovative k-means algorithm that utilizes end node characteristics based on the graph partitioning method and the Silhouette technique, transforming the existing system into an MMG framework. Furthermore, an iterative pareto-fuzzy (IPF) method is suggested to determine optimal RES and BESS sizes, considering reliability, total cost, and the unutilized surplus power function. The framework integrates total line losses into the generation adequacy assessment of energy management strategy (EMS) for each MG within the MMG system. The design and performance analysis of the proposed multi-objective optimization algorithm is tested on the IEEE 33-bus distribution system. The implementation of the independent operation index (IOI) and round-trip efficiency (RTE) metrics demonstrates the effectiveness of the proposed approach in constructing an MMG, with achieved values exceeding 90 percent for IOI and 77 percent for RTE.

Counting connected partitions of graphs

AbstractMotivated by the theorem of Győri and Lovász, we consider the following problem. For a connected graph on vertices and edges determine the number of unordered solutions of positive integers such that every is realized by a connected subgraph of with edges. We also consider the vertex‐partition analogue. We prove various lower bounds on as a function of the number of vertices in , as a function of the average degree of , and also as the size of ‐partite connected maximum cuts of . Those three lower bounds are tight up to a multiplicative constant. We also prove that the number of unordered ‐tuples with , that are realizable by vertex partitions into connected parts, is at least .

Fast Dynamic Time Warping for Temperature Compensation in Guided Waves

The paper discusses Structural Health Monitoring (SHM) based on ultrasonic guided waves for damage detection in structures. Guided waves allow inspection over long distances and inaccessible features, but are also sensitive to changes in environmental and operating conditions (EOC). The focus of this paper is on temperature compensation methods for guided waves. The compensation techniques include Baseline Signal Stretch (BSS), Optimal Baseline Selection (OBS), OBS+BSS, and Dynamic Time-Warping (DTW). In particular, a new, fast approximation of DTW is evaluated and compared with conventional but computationally expensive DTW. The FDTW algorithm utilizes a multilevel approach inspired by graph bisection principles to achieve precise warping path determination with linear computational complexity. The study evaluates the compensation performance of FDTW using a single baseline at a single temperature, thus addressing the complexity and inaccessibility issues of obtaining an extensive database of baseline signals in practical applications. The Open Guided Wave (OGW) dataset is employed for a fair comparison with other compensation methods. Results indicate that FDTW performs well, demonstrating comparable warping performance to DTW but with significantly reduced computational complexity. The analysis also includes comparisons with BSS, OBS+BSS, and DTW across a range of temperatures, highlighting the effectiveness of FDTW in mitigating errors introduced by larger temperature variations.

The equivalences of community detection methods for bipartite networks

Investigating the community structures of bipartite networks is a frequent topic of discussion in the ecological and social fields. The most widely used methods, as proposed by numerous academics from varying perspectives, include spectral graph partitioning, modularity, non-negative matrix factorization, and stochastic block model. In this paper, we demonstrate three equivalences among these four methods. i) Both Dhillon spectral graph partitioning and Barber modularity clustering are equivalent to solving for the matrix's left and right singular vectors after relaxing the discrete constraints. ii) The non-negative matrix factorization clustering is equivalent to the Dhillon spectral graph partitioning. iii) The bipartite stochastic block model is equivalent to the constraint-based NMF that uses K-L divergence as its cost function. These equivalences, obtained through rigorous mathematical derivations, will aid in the future development of efficient algorithms for community detection in bipartite networks.

Play like a Vertex: A Stackelberg Game Approach for Streaming Graph Partitioning

In the realm of distributed systems tasked with managing and processing large-scale graph-structured data, optimizing graph partitioning stands as a pivotal challenge. The primary goal is to minimize communication overhead and runtime cost. However, alongside the computational complexity associated with optimal graph partitioning, a critical factor to consider is memory overhead. Real-world graphs often reach colossal sizes, making it impractical and economically unviable to load the entire graph into memory for partitioning. This is also a fundamental premise in distributed graph processing, where accommodating a graph with non-distributed systems is unattainable. Currently, existing streaming partitioning algorithms exhibit a skew-oblivious nature, yielding satisfactory partitioning results exclusively for specific graph types. In this paper, we propose a novel streaming partitioning algorithm, the Skewness-aware Vertex-cut Partitioner (S5P ), designed to leverage the skewness characteristics of real graphs for achieving high-quality partitioning. S5P offers high partitioning quality by segregating the graph's edge set into two subsets, head and tail sets. Following processing by a skewness-aware clustering algorithm, these two subsets subsequently undergo a Stackelberg graph game. Our extensive evaluations conducted on substantial real-world and synthetic graphs demonstrate that, in all instances, the partitioning quality of S5P surpasses that of existing streaming partitioning algorithms, operating within the same load balance constraints. For example, S5P can bring up to a 51% improvement in partitioning quality compared to the top partitioner among the baselines. Lastly, we showcase that the implementation of S5P results in up to an 81% reduction in communication cost and a 130% increase in runtime efficiency for distributed graph processing tasks on PowerGraph.

Analysis of Combinational Circuit Failure Rate based on Graph Partitioning and Probabilistic Binomial Approach

Analysis of Combinational Circuit Failure Rate based on Graph Partitioning and Probabilistic Binomial Approach

Gain and Pain in Graph Partitioning: Finding Accurate Communities in Complex Networks

This paper presents an approach to community detection in complex networks by simultaneously incorporating a connectivity-based metric and Max-Min Modularity. By leveraging the connectivity-based metric and employing a heuristic algorithm, we develop a novel complementary graph for the Max-Min Modularity that enhances its effectiveness. We formulate community detection as an integer programming problem of an equivalent yet more compact counterpart model of the revised Max-Min Modularity maximization problem. Using a row generation technique alongside the heuristic approach, we then provide a hybrid procedure for near-optimally solving the model and discovering high-quality communities. Through a series of experiments, we demonstrate the success of our algorithm, showcasing its efficiency in detecting communities, particularly in extensive networks.

Partitioning graphs with linear minimum degree

AbstractWe prove that there exists an absolute constant such that, for any positive integer , every graph with minimum degree at least admits a vertex‐partition , where both and have minimum degree at least , and every vertex in has at least neighbors in . This confirms a question posted by Kühn and Osthus (J. Comb. Theory Ser. B. 88 (2003), 29–43.) and is tight up to a constant factor. Our proof combines probabilistic methods with structural arguments based on Ore's theorem on ‐factors of bipartite graphs.

Exactly solvable Hamiltonian fragments obtained from a direct sum of Lie algebras.

Exactly solvable Hamiltonians are useful in the study of quantum many-body systems using quantum computers. In the variational quantum eigensolver, a decomposition of the target Hamiltonian into exactly solvable fragments can be used for the evaluation of the energies via repeated quantum measurements. In this work, we apply more general classes of exactly solvable qubit Hamiltonians than previously considered to address the Hamiltonian measurement problem. The most general exactly solvable Hamiltonians we use are defined by the condition that within each simultaneous eigenspace of a set of Pauli symmetries, the Hamiltonian acts effectively as an element of a direct sum of so(N) Lie algebras and can, therefore, be measured using a combination of unitaries in the associated Lie group, Clifford unitaries, and mid-circuit measurements. The application of such Hamiltonians to decomposing molecular electronic Hamiltonians via graph partitioning techniques shows a reduction in the total number of measurements required to estimate the expectation value compared to previously used exactly solvable qubit Hamiltonians.

On minimum vertex bisection of random d-regular graphs

Minimum vertex bisection is a graph partitioning problem in which the aim is to find a partition of the vertices into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the other set. In this work we are interested in providing asymptotically almost surely upper bounds on the minimum vertex bisection of random d-regular graphs, for constant values of d. Our approach is based on analyzing a greedy algorithm by using the differential equation method. In this way, we obtain the first known non-trivial upper bounds for the vertex bisection number in random regular graphs. The numerical approximations of these theoretical bounds are compared with the emprical ones, and with the lower bounds from Kolesnik and Wormald (2014) [30].

Planar Graphs without Cycles of Length 3, 4, and 6 are (3, 3)-Colorable

For non-negative integers d1 and d2, if V1 and V2 are two partitions of a graph G’s vertex set VG, such that V1 and V2 induce two subgraphs of G, called GV1 with maximum degree at most d1 and GV2 with maximum degree at most d2, respectively, then the graph G is said to be improper d1,d2-colorable, as well as d1,d2-colorable. A class of planar graphs without C3,C4, and C6 is denoted by C. In 2019, Dross and Ochem proved that G is 0,6-colorable, for each graph G in C. Given that d1+d2≥6, this inspires us to investigate whether G is d1,d2-colorable, for each graph G in C. In this paper, we provide a partial solution by showing that G is (3, 3)-colorable, for each graph G in C.

Graph Partition and Multiple Choice-UCB Based Algorithms for Edge Server Placement in MEC Environment

Graph Partition and Multiple Choice-UCB Based Algorithms for Edge Server Placement in MEC Environment

Weak External Bisections of Regular Graphs

Weak External Bisections of Regular Graphs

FSM: A Fine-Grained Splitting and Merging Framework for Dual-Balanced Graph Partition

Partitioning a large graph into smaller subgraphs by minimizing the number of cutting vertices and edges, namely cut size or replication factor, plays a crucial role in distributed graph processing tasks. However, many prior works have primarily focused on optimizing the cut size by considering only vertex balance or edge balance, leading to significant workload imbalance and consequently hindering the performance of downstream tasks. Therefore, in this paper, we address the dual-balanced graph partition problem that minimizes the cut size while simultaneously guaranteeing both vertex and edge balance. We propose a lightweight effective two-phase framework, namely fine-grained splitting and merging (FSM), which decomposes the graph into more and smaller partitions and then merges them. FSM offers the flexibility of integrating with various state-of-the-art single-balanced techniques. We develop two efficient algorithms Fast Merging and Precise Merging to enable trade-offs between computational efficiency and partitioning quality. Experimental results on large real-world graphs demonstrate that FSM achieves state-of-the-art cut size while maintaining dual balance. The runtime for downstream tasks PageRank, connected component, and diameter estimation, can be reduced by a large proportion, up to 9.43%, 11.35%, and 17.94%, respectively.

FragQC: An efficient quantum error reduction technique using quantum circuit fragmentation

Quantum computers must meet extremely stringent qualitative and quantitative requirements on their qubits in order to solve real-life problems. Quantum circuit fragmentation techniques divide a large quantum circuit into a number of sub-circuits that can be executed on the smaller noisy quantum hardware available. However, the process of quantum circuit fragmentation involves finding an ideal cut that has exponential time complexity and also the classical post-processing required to reconstruct the output. In this paper, we represent a quantum circuit using a weighted graph and propose a novel classical graph partitioning algorithm for selecting an efficient fragmentation that reduces the entanglement between the sub-circuits along with balancing the estimated error in each sub-circuit. We also demonstrate a comparative study of different classical and quantum approaches to graph partitioning for finding such a cut. We present FragQC, a software tool that cuts a quantum circuit into sub-circuits when its error probability exceeds a certain threshold. With this proposed approach, we achieve an increase in fidelity of 13.38% compared to direct execution without cutting the circuit, and 7.88% over the state-of-the-art ILP-based method for the benchmark circuits.The code for FragQC is available at https://github.com/arnavdas88/FragQC.

GraphSER: Distance-Aware Stream-Based Edge Repartition for Many-Core Systems

With the explosive growth of graph data, distributed graph processing becomes popular and many graph hardware accelerators use distributed frameworks. Graph partitioning is foundation in distributed graph processing. However, dynamic changes in graph make existing partitioning shifted from its optimized points and cause system performance degraded. Therefore, more efficient dynamic graph partition methods are needed. In this work, we propose GraphSER, a dynamic graph partition method for many-core systems. In order to improve the cross-node spatial locality and reduce the overhead of repartition, we propose a stream-based edge repartition, in which each computing node sequentially traverses its local edge list in parallel, then migrating edges based on distance and replica degree. GraphSER does not need costly searching and prioritizes nodes so it can avoid poor cross-node spatial locality. Our evaluation shows that compared to state-of-the-art edge repartition software methods, GraphSER has an average speedup 1.52x, with the maximum up to 2x. Compared to the previous many-core hardware repartition method, GraphSER performance has an average of 40% improvement, with the maximum to 117%.