Scalable High-Quality Graph and Hypergraph Partitioning

The balanced hypergraph partitioning problem (HGP) asks for a partition of the node set of a hypergraph into $k$ blocks of roughly equal size, such that an objective function defined on the hyperedges is minimized. In this work, we optimize the connectivity metric, which is the most prominent objective function for HGP. The hypergraph partitioning problem is NP-hard and there exists no constant factor approximation. Thus, heuristic algorithms are used in practice with the multilevel scheme as the most successful approach to solve the problem: First, the input hypergraph is coarsened to obtain a hierarchy of successively smaller and structurally similar approximations. The smallest hypergraph is then initially partitioned into $k$ blocks, and subsequently, the contractions are reverted level-by-level, and, on each level, local search algorithms are used to improve the partition (refinement phase). In recent years, several new techniques were developed for sequential multilevel partitioning that substantially improved solution quality at the cost of an increased running time. These developments divide the landscape of existing partitioning algorithms into systems that either aim for speed or high solution quality with the former often being more than an order of magnitude faster than the latter. Due to the high running times of the best sequential algorithms, it is currently not feasible to partition the largest real-world hypergraphs with the highest possible quality. Thus, it becomes increasingly important to parallelize the techniques used in these algorithms. However, existing state-of-the-art parallel partitioners currently do not achieve the same solution quality as their sequential counterparts because they use comparatively weak components that are easier to parallelize. Moreover, there has been a recent trend toward simpler methods for partitioning large hypergraphs that even omit the multilevel scheme. In contrast to this development, we present two shared-memory multilevel hypergraph partitioners with parallel implementations of techniques used by the highest-quality sequential systems. Our first multilevel algorithm uses a parallel clustering-based coarsening scheme, which uses substantially fewer locking mechanisms than previous approaches. The contraction decisions are guided by the community structure of the input hypergraph obtained via a parallel community detection algorithm. For initial partitioning, we implement parallel multilevel recursive bipartitioning with a novel work-stealing approach and a portfolio of initial bipartitioning techniques to compute an initial solution. In the refinement phase, we use three different parallel improvement algorithms: label propagation refinement, a highly-localized direct $k$-way FM algorithm, and a novel parallelization of flow-based refinement. These algorithms build on our highly-engineered partition data structure, for which we propose several novel techniques to compute accurate gain values of node moves in the parallel setting. Our second multilevel algorithm parallelizes the $n$-level partitioning scheme used in the highest-quality sequential partitioner KaHyPar. Here, only a single node is contracted on each level, leading to a hierarchy with approximately $n$ levels where $n$ is the number of nodes. Correspondingly, in each refinement step, only a single node is uncontracted, allowing a highly-localized search for improvements. We show that this approach, which seems inherently sequential, can be parallelized efficiently without compromises in solution quality. To this end, we design a forest-based representation of contractions from which we derive a feasible parallel schedule of the contraction operations that we apply on a novel dynamic hypergraph data structure on-the-fly. In the uncoarsening phase, we decompose the contraction forest into batches, each containing a fixed number of nodes. We then uncontract each batch in parallel and use highly-localized versions of our refinement algorithms to improve the partition around the uncontracted nodes. We further show that existing sequential partitioning algorithms considerably struggle to find balanced partitions for weighted real-world hypergraphs. To this end, we present a technique that enables partitioners based on recursive bipartitioning to reliably compute balanced solutions. The idea is to preassign a small portion of the heaviest nodes to one of the two blocks of each bipartition and optimize the objective function on the remaining nodes. We integrated the approach into the sequential hypergraph partitioner KaHyPar and show that our new approach can compute balanced solutions for all tested instances without negatively affecting the solution quality and running time of KaHyPar. In our experimental evaluation, we compare our new shared-memory (hyper)graph partitioner Mt-KaHyPar to $25$ different graph and hypergraph partitioners on over $800$ (hyper)graphs with up to two billion edges/pins. The results indicate that already our fastest configuration outperforms almost all existing hypergraph partitioners with regards to both solution quality and running time. Our highest-quality configuration ($n$-level with flow-based refinement) achieves the same solution quality as the currently best sequential partitioner KaHyPar, while being almost an order of magnitude faster with ten threads. In addition, we optimize our data structures for graph partitioning, which improves the running times of both multilevel partitioners by almost a factor of two for graphs. As a result, Mt-KaHyPar also outperforms most of the existing graph partitioning algorithms. While the shared-memory graph partitioner KaMinPar is still faster than Mt-KaHyPar, its produced solutions are worse by $10\%$ in the median. The best sequential graph partitioner KaFFPa-StrongS computes slightly better partitions than Mt-KaHyPar (median improvement is $1\%$), but is more than an order of magnitude slower on average

Parallel and External High Quality Graph Partitioning

Partitioning graphs into k blocks of roughly equal size such that few edges run between the blocks is a key tool for processing and analyzing large complex real-world networks. The graph partitioning problem has multiple practical applications in parallel and distributed computations, data storage, image processing, VLSI physical design and many more. Furthermore, recently, size, variety, and structural complexity of real-world networks has grown dramatically. Therefore, there is a demand for efficient graph partitioning algorithms that fully utilize computational power and memory capacity of modern machines. A popular and successful heuristic to compute a high-quality partitions of large networks in reasonable time is $\textit{multi-level graph partitioning}$ approach which contracts the graph preserving its structure and then partitions it using a complex graph partitioning algorithm. Specifically, the multi-level graph partitioning approach consists of three main phases: coarsening, initial partitioning, and uncoarsening. During the coarsening phase, the graph is recursively contracted preserving its structure and properties until it is small enough to compute its initial partition during the initial partitioning phase. Afterwards, during the uncoarsening phase the partition of the contracted graph is projected onto the original graph and refined using, for example, local search. Most of the research on heuristical graph partitioning focuses on sequential algorithms or parallel algorithms in the distributed memory model. Unfortunately, previous approaches to graph partitioning are not able to process large networks and rarely take in into account several aspects of modern computational machines. Specifically, the amount of cores per chip grows each year as well as the price of RAM reduces slower than the real-world graphs grow. Since HDDs and SSDs are 50 – 400 times cheaper than RAM, external memory makes it possible to process large real-world graphs for a reasonable price. Therefore, in order to better utilize contemporary computational machines, we develop efficient $\textit{multi-level graph partitioning}$ algorithms for the shared-memory and the external memory models. First, we present an approach to shared-memory parallel multi-level graph partitioning that guarantees balanced solutions, shows high speed-ups for a variety of large graphs and yields very good quality independently of the number of cores used. Important ingredients include parallel label propagation for both coarsening and uncoarsening, parallel initial partitioning, a simple yet effective approach to parallel localized local search, and fast locality preserving hash tables that effectively utilizes caches. The main idea of the parallel localized local search is that each processors refines only a small area around a random vertex reducing interactions between processors. For example, on 79 cores, our algorithms partitions a graph with more than 3 billions of edges into 16 blocks cutting 4.5% less edges than the closest competitor and being more than two times faster. Furthermore, another competitors is not able to partition this graph. We then present an approach to external memory graph partitioning that is able to partition large graphs that do not fit into RAM. Specifically, we consider the semi-external and the external memory model. In both models a data structure of size proportional to the number of edges does not fit into the RAM. The difference is that the former model assumes that a data structure of size proportional to the number of vertices fits into the RAM whereas the latter assumes the opposite. We address the graph partitioning problem in both models by adapting the size-constrained label propagation technique for the semi-external model and by developing a size-constrained clustering algorithm based on graph coloring in the external memory. Our semi-external size-constrained label propagation algorithm (or external memory clustering algorithm) can be used to compute graph clusterings and is a prerequisite for the (semi-)external graph partitioning algorithm. The algorithms are then used for both the coarsening and the uncoarsening phase of a multi-level algorithm to compute graph partitions. Our (semi-)external algorithm is able to partition and cluster huge complex networks with billions of edges on cheap commodity machines. Experiments demonstrate that the semi-external graph partitioning algorithm is scalable and can compute high quality partitions in time that is comparable to the running time of an efficient internal memory implementation. A parallelization of the algorithm in the semi-external model further reduces running times. Additionally, we develop a speed-up technique for the hypergraph partitioning algorithms. Hypergraphs are an extension of graphs that allow a single edge to connect more than two vertices. Therefore, they describe models and processes more accurately additionally allowing more possibilities for improvement. Most multi-level hypergraph partitioning algorithms perform some computations on vertices and their set of neighbors. Since these computations can be super-linear, they have a significant impact on the overall running time on large hypergraphs. Therefore, to further reduce the size of hyperedges, we develop a pin-sparsifier based on the min-hash technique that clusters vertices with similar neighborhood. Further, vertices that belong to the same cluster are substituted by one vertex, which is connected to their neighbors, therefore, reducing the size of the hypergraph. Our algorithm sparsifies a hypergraph such that the resulting graph can be partitioned significantly faster without loss in quality (or with insignificant loss). On average, KaHyPar with sparsifier performs partitioning about 1.5 times faster while preserving solution quality if hyperedges are large. All aforementioned frameworks are publicly available

High-Quality Shared-Memory Graph Partitioning

Partitioning graphs into blocks of roughly equal size such that few edges run between blocks is a frequently needed operation in processing graphs. Recently, size, variety, and structural complexity of these networks has grown dramatically. Unfortunately, previous approaches to parallel graph partitioning have problems in this context since they often show a negative trade-off between speed and quality. We present an approach to multi-level shared-memory parallel graph partitioning that guarantees balanced solutions, shows high speed-ups for a variety of large graphs and yields very good quality independently of the number of cores used. For example, on 31 cores, our algorithm partitions our largest test instance into 16 blocks cutting less than half the number of edges than our main competitor when both algorithms are given the same amount of time. Important ingredients include parallel label propagation for both coarsening and improvement, parallel initial partitioning, a simple yet effective approach to parallel localized local search, and fast locality preserving hash tables

Parallel Graph Partitioning for Complex Networks

Processing large complex networks like social networks or web graphs has recently attracted considerable interest. In order to do this in parallel, we need to partition them into pieces of about equal size. Unfortunately, previous parallel graph partitioners originally developed for more regular mesh-like networks do not work well for these networks. This paper addresses this problem by parallelizing and adapting the label propagation technique originally developed for graph clustering. By introducing size constraints, label propagation becomes applicable for both the coarsening and the refinement phase of multilevel graph partitioning. We obtain very high quality by applying a highly parallel evolutionary algorithm to the coarsened graph. The resulting system is both more scalable and achieves higher quality than state-of-the-art systems like ParMetis or PT-Scotch. For large complex networks the performance differences are very big. For example, our algorithm can partition a web graph with 3.3 billion edges in less than sixteen seconds using 512 cores of a high performance cluster while producing a high quality partition -- none of the competing systems can handle this graph on our system.Comment: Review article. Parallelization of our previous approach arXiv:1402.328

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Memetic Multilevel Hypergraph Partitioning

Hypergraph partitioning has a wide range of important applications such as VLSI design or scientific computing. With focus on solution quality, we develop the first multilevel memetic algorithm to tackle the problem. Key components of our contribution are new effective multilevel recombination and mutation operations that provide a large amount of diversity. We perform a wide range of experiments on a benchmark set containing instances from application areas such VLSI, SAT solving, social networks, and scientific computing. Compared to the state-of-the-art hypergraph partitioning tools hMetis, PaToH, and KaHyPar, our new algorithm computes the best result on almost all instances