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

Improving Coarsening Schemes for Hypergraph Partitioning by Exploiting Community Structure

We present an improved coarsening process for multilevel hypergraph partitioning that incorporates global information about the community structure. Community detection is performed via modularity maximization on a bipartite graph representation. The approach is made suitable for different classes of hypergraphs by defining weights for the graph edges that express structural properties of the hypergraph. We integrate our approach into a leading multilevel hypergraph partitioner with strong local search algorithms and perform extensive experiments on a large benchmark set of hypergraphs stemming from application areas such as VLSI design, SAT solving, and scientific computing. Our results indicate that respecting community structure during coarsening not only significantly improves the solutions found by the initial partitioning algorithm, but also consistently improves overall solution quality

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

Recent Advances in Graph Partitioning

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

Partitioning a given circuit targeting multiple Fpgas

Our approach to the problem of partitioning the design (represented as a hypergraph) into Multi-FPGAs uses a bi-level approach by initially clustering the design and then applying the bipartitioning technique iteratively. Each partition generated by the iterative bipartitioning technique should meet the constraints given by the FPGAs input-output and number of CLBs. The traditional FM partitioning can be applied to partition the circuit into multiple FPGAs. FM partitioning aims to minimize the number of interconnections but fails to group the nodes with maximum interconnections into one partition. Thus FM algorithm looks at the partitioning problem with a global viewpoint, abandoning the details. The proposed algorithm adds another level of optimization to the partitioning heuristic. By clustering the nodes that are connected very closely in a netlist before partitioning, local optimization property is added to the FM algorithm. This clustered circuit is then partitioned to implement the design in multiple FPGAs. Bipartitioning using the Fiduccia Mattheyses algorithm is applied. (Abstract shortened by UMI.)

Beyond pairwise clustering

We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a two-step algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms

Relaxation-Based Coarsening for Multilevel Hypergraph Partitioning

Multilevel partitioning methods that are inspired by principles of multiscaling are the most powerful practical hypergraph partitioning solvers. Hypergraph partitioning has many applications in disciplines ranging from scientific computing to data science. In this paper we introduce the concept of algebraic distance on hypergraphs and demonstrate its use as an algorithmic component in the coarsening stage of multilevel hypergraph partitioning solvers. The algebraic distance is a vertex distance measure that extends hyperedge weights for capturing the local connectivity of vertices which is critical for hypergraph coarsening schemes. The practical effectiveness of the proposed measure and corresponding coarsening scheme is demonstrated through extensive computational experiments on a diverse set of problems. Finally, we propose a benchmark of hypergraph partitioning problems to compare the quality of other solvers