Advanced Flow-Based Multilevel Hypergraph Partitioning

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 hyperedges are aggregated during the coarsening phase, refinement algorithms employed in the multilevel setting must be able to handle both weighted hyperedges and weighted vertices - even if the initial input hypergraph is unweighted. We therefore enhance HyperFlowCutter to handle weighted instances and propose a technique for computing maximum flows directly on weighted hypergraphs. We compare the performance of two configurations of our new algorithm with KaHyPar-MF and seven other partitioning algorithms on a comprehensive benchmark set with instances from application areas such as VLSI design, scientific computing, and SAT solving. Our first configuration, KaHyPar-HFC, computes slightly better solutions than KaHyPar-MF using significantly less running time. The second configuration, KaHyPar-HFC*, computes solutions of significantly better quality and is still slightly faster than KaHyPar-MF. Furthermore, in terms of solution quality, both configurations also outperform all other competing partitioners

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

Recent Advances in Graph Partitioning

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

Network Flow-Based Refinement for Multilevel Hypergraph Partitioning

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 from different application areas while still having a running time comparable to that of hMetis

Balanced Coarsening for Multilevel Hypergraph Partitioning via Wasserstein Discrepancy

We propose a balanced coarsening scheme for multilevel hypergraph partitioning. In addition, an initial partitioning algorithm is designed to improve the quality of k-way hypergraph partitioning. By assigning vertex weights through the LPT algorithm, we generate a prior hypergraph under a relaxed balance constraint. With the prior hypergraph, we have defined the Wasserstein discrepancy to coordinate the optimal transport of coarsening process. And the optimal transport matrix is solved by Sinkhorn algorithm. Our coarsening scheme fully takes into account the minimization of connectivity metric (objective function). For the initial partitioning stage, we define a normalized cut function induced by Fiedler vector, which is theoretically proved to be a concave function. Thereby, a three-point algorithm is designed to find the best cut under the balance constraint