A directed acyclic hypergraph is a generalized concept of a directed acyclic graph, where each hyperedge can contain an arbitrary number of tails and heads. Directed hypergraphs can be used to model data flow and execution dependencies in streaming applications. Thus, hypergraph partitioning algorithms can be used to obtain efficient parallelizations for multiprocessor architectures. However, an acyclicity constraint on the partition is necessary when mapping streaming applications to embedded multiprocessors due to resource restrictions on this type of hardware. The acyclic hypergraph partitioning problem is to partition the hypernodes of a directed acyclic hypergraph into a given number of blocks of roughly equal size such that the corresponding quotient graph is acyclic while minimizing an objective function on the partition.
Here, we contribute the first n-level algorithm for the acyclic hypergraph partitioning problem. Our focus is on acyclic hypergraphs where hyperedges can have one head and arbitrary many tails. Based on this, we engineer a memetic algorithm to further reduce communication cost, as well as to improve scheduling makespan on embedded multiprocessor architectures. Experiments indicate that our algorithm outperforms previous algorithms that focus on the directed acyclic graph case which have previously been employed in the application domain. Moreover, our experiments indicate that using the directed hypergraph model for this type of application yields a significantly smaller makespan.
We describe the engineering of the distributed-memory multilevel graph partitioner dKaMinPar. It scales to (at least) 8192 cores while achieving partitioning quality comparable to widely used sequential and shared-memory graph partitioners. In comparison, previous distributed graph partitioners scale only in more restricted scenarios and often induce a considerable quality penalty compared to non-distributed partitioners. When partitioning into a large number of blocks, they even produce infeasible solution that violate the balancing constraint. dKaMinPar achieves its robustness by a scalable …
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 computation for multiprocessor architectures. However due to resource restrictions, an acyclicity constraint on the partition is necessary when mapping streaming applications to an embedded multiprocessor. Here, we contribute a multi-level algorithm for the acyclic graph partitioning problem. Based on this, we engineer an evolutionary algorithm to further reduce communication cost, as well as to i…
We present a multi-level graph partitioning algorithm using novel local improvement algorithms and global search strategies transferred from the multi-grid community. Local improvement algorithms are based max-flow min-cut computations and more localized FM searches. By combining these techniques, we obtain an algorithm that is fast on the one hand and on the other hand is able to improve the best known partitioning results for many inputs. For example, in Walshaw's well known benchmark tables we achieve 317 improvements for the tables 1%, 3% and 5% imbalance. Moreover, in 118 additional cases…
Algorithms for many hypergraph problems, including partitioning, utilize multilevel frameworks to achieve a good trade-off between the performance and the quality of results. In this paper we introduce two novel aggregative coarsening schemes and incorporate them within state-of-the-art hypergraph partitioner Zoltan. Our coarsening schemes are inspired by the algebraic multigrid and stable matching approaches. We demonstrate the effectiveness of the developed schemes as a part of multilevel hypergraph partitioning framework on a wide range of problems.
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,…
We present a refinement framework for multilevel hypergraph partitioning that uses max-flow computations on pairs of blocks to improve the solution quality of a $k$-way partition. The framework generalizes the flow-based improvement algorithm of KaFFPa from graphs to hypergraphs and is integrated into the hypergraph partitioner KaHyPar. By reducing the size of hypergraph flow networks, improving the flow model used in KaFFPa, and developing techniques to improve the running time of our algorithm, we obtain a partitioner that computes the best solutions for a wide range of benchmark hypergraphs…
Partitioning a graph into blocks of "roughly equal" weight while cutting only few edges is a fundamental problem in computer science with a wide range of applications. In particular, the problem is a building block in applications that require parallel processing. While the amount of available cores in parallel architectures has significantly increased in recent years, state-of-the-art graph partitioning algorithms do not work well if the input needs to be partitioned into a large number of blocks. Often currently available algorithms compute highly imbalanced solutions, solutions of low quali…
This work addresses the NP-Hard problem of acyclic directed acyclic graph (DAG) partitioning problem. The acyclic partitioning problem is defined as partitioning the vertex set of a given directed acyclic graph into disjoint and collectively exhaustive subsets (parts). Parts are to be assigned such that the total sum of the vertex weights within each part satisfies a common upper bound and the total sum of the edge costs that connect nodes across different parts is minimized. Additionally, the quotient graph, i.e., the induced graph where all nodes that are assigned to the same part are contra…
Intel Corporation（英特尔公司）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；University of Vienna（维也纳大学）
Graphs are widely used to model execution dependencies in applications. In particular, the NP-complete problem of partitioning a graph under constraints receives enormous attention by researchers because of its applicability in multiprocessor scheduling. We identified the additional constraint of acyclic dependencies between blocks when mapping computer vision and imaging applications to a heterogeneous embedded multiprocessor. Existing algorithms and heuristics do not address this requirement and deliver results that are not applicable for our use-case. In this work, we show that this more co…
The graph partitioning problem is widely used and studied in many practical and theoretical applications. The multilevel strategies represent today one of the most effective and efficient generic frameworks for solving this problem on large-scale graphs. Most of the attention in designing the multilevel partitioning frameworks has been on the refinement phase. In this work we focus on the coarsening phase, which is responsible for creating structurally similar to the original but smaller graphs. We compare different matching- and AMG-based coarsening schemes, experiment with the algebraic dist…
The balanced hypergraph partitioning problem is to partition a hypergraph into $k$ disjoint blocks of bounded size such that the sum of the number of blocks connected by each hyperedge is minimized. We present an improvement to the flow-based refinement framework of KaHyPar-MF, the current state-of-the-art multilevel $k$-way hypergraph partitioning algorithm for high-quality solutions. Our improvement is based on the recently proposed HyperFlowCutter algorithm for computing bipartitions of unweighted hypergraphs by solving a sequence of incremental maximum flow problems. Since vertices and hyp…
The graph partitioning problem has many applications in scientific computing such as computer aided design, data mining, image compression and other applications with sparse-matrix vector multiplications as a kernel operation. In many cases it is advantageous to use hypergraphs as they, compared to graphs, have a more general structure and can be used to model more complex relationships between groups of objects. This motivates our focus on the less-studied hypergraph partitioning problem. In this paper, we propose a serial multi-level bipartitioning algorithm. One important step in current he…
The balanced hypergraph partitioning problem (HGP) is to partition the vertex set of a hypergraph into k disjoint blocks of bounded weight, while minimizing an objective function defined on the hyperedges. Whereas real-world applications often use vertex and edge weights to accurately model the underlying problem, the HGP research community commonly works with unweighted instances. In this paper, we argue that, in the presence of vertex weights, current balance constraint definitions either yield infeasible partitioning problems or allow unnecessarily large imbalances and propose a new definit…
This paper severs as a user guide to the graph partitioning framework KaHIP (Karlsruhe High Quality Partitioning). We give a rough overview of the techniques used within the framework and describe the user interface as well as the file formats used. Moreover, we provide a short description of the current library functions provided within the framework. Since version 3.00 we support multilevel partitioning, memetic algorithms, distributed and shared-memory parallel algorithms, node separator and ordering algorithms, edge partitioning algorithms as well as ILP solvers.
Partitioning graphs into blocks of roughly equal size such that few edges run between blocks is a frequently needed operation when processing graphs on a parallel computer. When a topology of a distributed system is known an important task is then to map the blocks of the partition onto the processors such that the overall communication cost is reduced. We present novel multilevel algorithms that integrate graph partitioning and process mapping. Important ingredients of our algorithm include fast label propagation, more localized local search, initial partitioning, as well as a compressed data…
Hypergraph partitioning is an important preprocessing step for optimizing data placement and minimizing communication volumes in high-performance computing applications. To cope with ever growing problem sizes, it has become increasingly important to develop fast parallel partitioning algorithms whose solution quality is competitive with existing sequential algorithms. To this end, we present Mt-KaHyPar, the first shared-memory multilevel hypergraph partitioner with parallel implementations of many techniques used by the sequential, high-quality partitioning systems: a parallel coarsening algo…
We present a shared-memory algorithm to compute high-quality solutions to the balanced $k$-way hypergraph partitioning problem. This problem asks for a partition of the vertex set into $k$ disjoint blocks of bounded size that minimizes the connectivity metric (i.e., the sum of the number of different blocks connected by each hyperedge). High solution quality is achieved by parallelizing the core technique of the currently best sequential partitioner KaHyPar: the most extreme $n$-level version of the widely used multilevel paradigm, where only a single vertex is contracted on each level. This a…
We develop a multilevel algorithm for hypergraph partitioning that contracts the vertices one at a time. Using several caching and lazy-evaluation techniques during coarsening and refinement, we reduce the running time by up to two-orders of magnitude compared to a naive $n$-level algorithm that would be adequate for ordinary graph partitioning. The overall performance is even better than the widely used hMetis hypergraph partitioner that uses a classical multilevel algorithm with few levels. Aided by a portfolio-based approach to initial partitioning and adaptive budgeting of imbalance within…
In this article we will present a graph partitioning algorithm which partitions a graph into two different types of components: the well-known `strongly connected components' as well as another type of components we call `connected acyclic component'. We will give an algorithm based on Tarjan's algorithm for finding strongly connected components used to find such a partitioning. We will also show that the partitioning given by the algorithm is unique and that the underlying graph can be represented as a directed acyclic graph (similar to a pure strongly connected component partitioning). In th…
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 t…
Many applications produce massive complex networks whose analysis would benefit from parallel processing. Parallel algorithms, in turn, often require a suitable network partition. For solving optimization tasks such as graph partitioning on large networks, multilevel methods are preferred in practice. Yet, complex networks pose challenges to established multilevel algorithms, in particular to their coarsening phase. One way to specify a (recursive) coarsening of a graph is to rate its edges and then contract the edges as prioritized by the rating. In this paper we (i) define weights for the ed…
This paper proposes multiple extensions to the popular bicriterion transit routing approach -- Trip-Based Transit Routing (TBTR). Specifically, building on the premise of the HypRAPTOR algorithm, we first extend TBTR to its partitioning variant -- HypTBTR. However, the improvement in query times of HyTBTR over TBTR comes at the cost of increased preprocessing. To counter this issue, two new techniques are proposed -- a One-To-Many variant of TBTR and multilevel partitioning. Our One-To-Many algorithm can rapidly solve profile queries, which not only reduces the preprocessing time for HypTBTR, …
Partitioning graphs into blocks of roughly equal size is widely used when processing large graphs. Currently there is a gap in the space of available partitioning algorithms. On the one hand, there are streaming algorithms that have been adopted to partition massive graph data on small machines. In the streaming model, vertices arrive one at a time including their neighborhood and then have to be assigned directly to a block. These algorithms can partition huge graphs quickly with little memory, but they produce partitions with low solution quality. On the other hand, there are offline (shared…
The polynomial partitioning method of Guth and Katz [arXiv:1011.4105] has numerous applications in discrete and computational geometry. It partitions a given $n$-point set $P\subset\mathbb{R}^d$ using the zero set $Z(f)$ of a suitable $d$-variate polynomial $f$. Applications of this result are often complicated by the problem, what should be done with the points of $P$ lying within $Z(f)$? A natural approach is to partition these points with another polynomial and continue further in a similar manner. So far it has been pursued with limited success---several authors managed to construct and ap…
In this paper we introduce the first efficient external-memory algorithm to compute the bisimilarity equivalence classes of a directed acyclic graph (DAG). DAGs are commonly used to model data in a wide variety of practical applications, ranging from XML documents and data provenance models, to web taxonomies and scientific workflows. In the study of efficient reasoning over massive graphs, the notion of node bisimilarity plays a central role. For example, grouping together bisimilar nodes in an XML data set is the first step in many sophisticated approaches to building indexing data structure…
We present a shared-memory parallelization of flow-based refinement, which is considered the most powerful iterative improvement technique for hypergraph partitioning at the moment. Flow-based refinement works on bipartitions, so current sequential partitioners schedule it on different block pairs to improve $k$-way partitions. We investigate two different sources of parallelism: a parallel scheduling scheme and a parallel maximum flow algorithm based on the well-known push-relabel algorithm. In addition to thoroughly engineered implementations, we propose several optimizations that substantia…
A node separator of a graph is a subset S of the nodes such that removing S and its incident edges divides the graph into two disconnected components of about equal size. In this work, we introduce novel algorithms to find small node separators in large graphs. With focus on solution quality, we introduce novel flow-based local search algorithms which are integrated in a multilevel framework. In addition, we transfer techniques successfully used in the graph partitioning field. This includes the usage of edge ratings tailored to our problem to guide the graph coarsening algorithm as well as hi…
We develop a multilevel algorithm for hypergraph partitioning that contracts the vertices one at a time and thus allows very high quality. This includes a rating function that avoids nonuniform vertex weights, an efficient "semi-dynamic" hypergraph data structure, a very fast coarsening algorithm, and two new local search algorithms. One is a $k$-way hypergraph adaptation of Fiduccia-Mattheyses local search and gives high quality at reasonable cost. The other is an adaptation of size-constrained label propagation to hypergraphs. Comparisons with hMetis and PaToH indicate that the new algorithm…
In this paper, we develop semi-external and external memory algorithms for graph partitioning and clustering problems. Graph partitioning and clustering are key tools for processing and analyzing large complex networks. We address both problems in the (semi-)external model by adapting the size-constrained label propagation technique. Our (semi-)external size-constrained label propagation algorithm can be used to compute graph clusterings and is a prerequisite for the (semi-)external graph partitioning algorithm. The algorithm is then used for both the coarsening and the refinement phase of a m…
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 refine…
Emerging quantum processors provide an opportunity to explore new approaches for solving traditional problems in the post Moore's law supercomputing era. However, the limited number of qubits makes it infeasible to tackle massive real-world datasets directly in the near future, leading to new challenges in utilizing these quantum processors for practical purposes. Hybrid quantum-classical algorithms that leverage both quantum and classical types of devices are considered as one of the main strategies to apply quantum computing to large-scale problems. In this paper, we advocate the use of mult…
Problems in scientific computing, such as distributing large sparse matrix operations, have analogous formulations as hypergraph partitioning problems. A hypergraph is a generalization of a traditional graph wherein "hyperedges" may connect any number of nodes. As a result, hypergraph partitioning is an NP-Hard problem to both solve or approximate. State-of-the-art algorithms that solve this problem follow the multilevel paradigm, which begins by iteratively "coarsening" the input hypergraph to smaller problem instances that share key structural features. Once identifying an approximate proble…
City University of Hong Kong（香港城市大学）；Universite Catholique de Louvain（天主教鲁汶大学）
We propose two spectral algorithms for partitioning nodes in directed graphs respectively with a cyclic and an acyclic pattern of connection between groups of nodes. Our methods are based on the computation of extremal eigenvalues of the transition matrix associated to the directed graph. The two algorithms outperform state-of-the art methods for directed graph clustering on synthetic datasets, including methods based on blockmodels, bibliometric symmetrization and random walks. Our algorithms have the same space complexity as classical spectral clustering algorithms for undirected graphs and …
A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by …
Finding the origin of short phrases propagating through the web has been formalized by Leskovec et al. [ACM SIGKDD 2009] as DAG Partitioning: given an arc-weighted directed acyclic graph on $n$ vertices and $m$ arcs, delete arcs with total weight at most $k$ such that each resulting weakly-connected component contains exactly one sink---a vertex without outgoing arcs. DAG Partitioning is NP-hard. We show an algorithm to solve DAG Partitioning in $O(2^k \cdot (n+m))$ time, that is, in linear time for fixed $k$. We complement it with linear-time executable data reduction rules. Our experiments s…
In this paper, we propose HyperFlowCutter, an algorithm for balanced hypergraph bipartitioning. It is based on minimum S-T hyperedge cuts and maximum flows. It computes a sequence of bipartitions that optimize cut size and balance in the Pareto sense, being able to trade one for the other. HyperFlowCutter builds on the FlowCutter algorithm for partitioning graphs. We propose additional features, such as handling disconnected hypergraphs, novel methods for obtaining starting S,T pairs as well as an approach to refine a given partition with HyperFlowCutter. Our main contribution is ReBaHFC, a ne…
A tournament is a directed graph T such that every pair of vertices are connected by an arc. A feedback vertex set is a set S of vertices in T such that T - S is acyclic. In this article we consider the Feedback Vertex Set problem in tournaments. Here input is a tournament T and integer k, and the task is to determine whether T has a feedback vertex set of size at most k. We give a new algorithm for Feedback Vertex Set in Tournaments. The running time of our algorithm is upper bounded by O(1.618^k + n^{O(1)}) and by O(1.46^n). Thus our algorithm simultaneously improves over the fastest known p…
