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

High-Quality Hypergraph Partitioning

This dissertation focuses on computing high-quality solutions for the NP-hard balanced hypergraph partitioning problem: Given a hypergraph and an integer $k$, partition its vertex set into $k$ disjoint blocks of bounded size, while minimizing an objective function over the hyperedges. Here, we consider the two most commonly used objectives: the cut-net metric and the connectivity metric. Since the problem is computationally intractable, heuristics are used in practice - the most prominent being the three-phase multi-level paradigm: During coarsening, the hypergraph is successively contracted to obtain a hierarchy of smaller instances. After applying an initial partitioning algorithm to the smallest hypergraph, contraction is undone and, at each level, refinement algorithms try to improve the current solution. With this work, we give a brief overview of the field and present several algorithmic improvements to the multi-level paradigm. Instead of using a logarithmic number of levels like traditional algorithms, we present two coarsening algorithms that create a hierarchy of (nearly) $n$ levels, where $n$ is the number of vertices. This makes consecutive levels as similar as possible and provides many opportunities for refinement algorithms to improve the partition. This approach is made feasible in practice by tailoring all algorithms and data structures to the $n$-level paradigm, and developing lazy-evaluation techniques, caching mechanisms and early stopping criteria to speed up the partitioning process. Furthermore, we propose a sparsification algorithm based on locality-sensitive hashing that improves the running time for hypergraphs with large hyperedges, and show that incorporating global information about the community structure into the coarsening process improves quality. Moreover, we present a portfolio-based initial partitioning approach, and propose three refinement algorithms. Two are based on the Fiduccia-Mattheyses (FM) heuristic, but perform a highly localized search at each level. While one is designed for two-way partitioning, the other is the first FM-style algorithm that can be efficiently employed in the multi-level setting to directly improve $k$-way partitions. The third algorithm uses max-flow computations on pairs of blocks to refine $k$-way partitions. Finally, we present the first memetic multi-level hypergraph partitioning algorithm for an extensive exploration of the global solution space. All contributions are made available through our open-source framework KaHyPar. In a comprehensive experimental study, we compare KaHyPar with hMETIS, PaToH, Mondriaan, Zoltan-AlgD, and HYPE on a wide range of hypergraphs from several application areas. Our results indicate that KaHyPar, already without the memetic component, computes better solutions than all competing algorithms for both the cut-net and the connectivity metric, while being faster than Zoltan-AlgD and equally fast as hMETIS. Moreover, KaHyPar compares favorably with the current best graph partitioning system KaFFPa - both in terms of solution quality and running time

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

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

Partitioning Complex Networks via Size-constrained Clustering

The most commonly used method to tackle the graph partitioning problem in practice is the multilevel approach. During a coarsening phase, a multilevel graph partitioning algorithm reduces the graph size by iteratively contracting nodes and edges until the graph is small enough to be partitioned by some other algorithm. A partition of the input graph is then constructed by successively transferring the solution to the next finer graph and applying a local search algorithm to improve the current solution. In this paper, we describe a novel approach to partition graphs effectively especially if the networks have a highly irregular structure. More precisely, our algorithm provides graph coarsening by iteratively contracting size-constrained clusterings that are computed using a label propagation algorithm. The same algorithm that provides the size-constrained clusterings can also be used during uncoarsening as a fast and simple local search algorithm. Depending on the algorithm's configuration, we are able to compute partitions of very high quality outperforming all competitors, or partitions that are comparable to the best competitor in terms of quality, hMetis, while being nearly an order of magnitude faster on average. The fastest configuration partitions the largest graph available to us with 3.3 billion edges using a single machine in about ten minutes while cutting less than half of the edges than the fastest competitor, kMetis

Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures

Quantum computers have recently made great strides and are on a long-term path towards useful fault-tolerant computation. A dominant overhead in fault-tolerant quantum computation is the production of high-fidelity encoded qubits, called magic states, which enable reliable error-corrected computation. We present the first detailed designs of hardware functional units that implement space-time optimized magic-state factories for surface code error-corrected machines. Interactions among distant qubits require surface code braids (physical pathways on chip) which must be routed. Magic-state factories are circuits comprised of a complex set of braids that is more difficult to route than quantum circuits considered in previous work [1]. This paper explores the impact of scheduling techniques, such as gate reordering and qubit renaming, and we propose two novel mapping techniques: braid repulsion and dipole moment braid rotation. We combine these techniques with graph partitioning and community detection algorithms, and further introduce a stitching algorithm for mapping subgraphs onto a physical machine. Our results show a factor of 5.64 reduction in space-time volume compared to the best-known previous designs for magic-state factories.Comment: 13 pages, 10 figure