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.

[1]  Sebastian Schlag,et al.  High-Quality Hypergraph Partitioning , 2020 .

[2]  Sebastian Schlag,et al.  Multilevel Hypergraph Partitioning with Vertex Weights Revisited , 2020 .

[3]  Shaoyi Du,et al.  Hypergraph Learning: Methods and Practices , 2020, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  Regina Barzilay,et al.  Optimal Transport Graph Neural Networks , 2020, ArXiv.

[5]  Samir Chowdhury,et al.  Generalized Spectral Clustering via Gromov-Wasserstein Learning , 2020, AISTATS.

[6]  T. Abdelzaher,et al.  Hypergraph Learning with Line Expansion , 2020, ArXiv.

[7]  Stojce Nakov,et al.  Design and Implementation of a Parallel Markowitz Threshold Algorithm , 2020, SIAM J. Matrix Anal. Appl..

[8]  Sebastian Schlag,et al.  Advanced Flow-Based Multilevel Hypergraph Partitioning , 2020, SEA.

[9]  Karsten M. Borgwardt,et al.  Wasserstein Weisfeiler-Lehman Graph Kernels , 2019, NeurIPS.

[10]  Lawrence Carin,et al.  Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching , 2019, NeurIPS.

[11]  Hongyuan Zha,et al.  Gromov-Wasserstein Learning for Graph Matching and Node Embedding , 2019, ICML.

[12]  Peter Sanders,et al.  Network Flow-Based Refinement for Multilevel Hypergraph Partitioning , 2018, SEA.

[13]  Federico Della Croce,et al.  Longest Processing Time rule for identical parallel machines revisited , 2018, ArXiv.

[14]  Christian Schulz,et al.  Memetic multilevel hypergraph partitioning , 2017, GECCO.

[15]  Vladimir G. Kim,et al.  Entropic metric alignment for correspondence problems , 2016, ACM Trans. Graph..

[16]  Gabriel Peyré,et al.  Gromov-Wasserstein Averaging of Kernel and Distance Matrices , 2016, ICML.

[17]  Peter Sanders,et al.  k-way Hypergraph Partitioning via n-Level Recursive Bisection , 2015, ALENEX.

[18]  Jure Leskovec,et al.  {SNAP Datasets}: {Stanford} Large Network Dataset Collection , 2014 .

[19]  Peter Sanders,et al.  Partitioning Complex Networks via Size-Constrained Clustering , 2014, SEA.

[20]  Marco Cuturi,et al.  Sinkhorn Distances: Lightspeed Computation of Optimal Transport , 2013, NIPS.

[21]  Natarajan Viswanathan,et al.  The DAC 2012 routability-driven placement contest and benchmark suite , 2012, DAC Design Automation Conference 2012.

[22]  Uwe Naumann,et al.  Combinatorial Scientific Computing , 2012 .

[23]  Facundo Mémoli,et al.  Gromov–Wasserstein Distances and the Metric Approach to Object Matching , 2011, Found. Comput. Math..

[24]  Jieping Ye,et al.  Hypergraph spectral learning for multi-label classification , 2008, KDD.

[25]  William J. Knottenbelt,et al.  Parallel multilevel algorithms for hypergraph partitioning , 2008, J. Parallel Distributed Comput..

[26]  C.J.H. Mann,et al.  Handbook of Approximation: Algorithms and Metaheuristics , 2008 .

[27]  Rob H. Bisseling,et al.  Parallel hypergraph partitioning for scientific computing , 2006, Proceedings 20th IEEE International Parallel & Distributed Processing Symposium.

[28]  Chris Walshaw,et al.  A Combined Evolutionary Search and Multilevel Optimisation Approach to Graph-Partitioning , 2004, J. Glob. Optim..

[29]  Ümit V. Çatalyürek,et al.  Hypergraph-Partitioning-Based Decomposition for Parallel Sparse-Matrix Vector Multiplication , 1999, IEEE Trans. Parallel Distributed Syst..

[30]  G. Karypis,et al.  Multilevel k-way hypergraph partitioning , 1999, Proceedings 1999 Design Automation Conference (Cat. No. 99CH36361).

[31]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[32]  Charles J. Alpert,et al.  The ISPD98 circuit benchmark suite , 1998, ISPD '98.

[33]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[34]  S. Shekhar,et al.  Multilevel Hypergraph Partitioning: Application In Vlsi Domain , 1997, Proceedings of the 34th Design Automation Conference.

[35]  Dennis J.-H. Huang,et al.  Multilevel Circuit Partitioning , 1997, Proceedings of the 34th Design Automation Conference.

[36]  Martine D. F. Schlag,et al.  Multi-level spectral hypergraph partitioning with arbitrary vertex sizes , 1996, Proceedings of International Conference on Computer Aided Design.

[37]  Ronald L. Graham,et al.  Bounds on Multiprocessing Timing Anomalies , 1969, SIAM Journal of Applied Mathematics.

[38]  Peter Sanders,et al.  Engineering a direct k-way Hypergraph Partitioning Algorithm , 2017, ALENEX.

[39]  Sebastian Schlag,et al.  Improving Coarsening Schemes for Hypergraph Partitioning by Exploiting Community Structure , 2017, SEA.

[40]  Ümit V. Çatalyürek,et al.  PaToH: Partitioning Tool for Hypergraphs , 1999 .

[41]  Igor L. Markov,et al.  Hypergraph Partitioning and Clustering , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[42]  Brendan Vastenhouw,et al.  A Two-Dimensional Data Distribution Method for Parallel Sparse Matrix-Vector Multiplication , 2005, SIAM Rev..

[43]  M. Fiedler.  Laplacian of graphs and algebraic connectivity , 1989 .

[44]  R. M. Mattheyses,et al.  A Linear-Time Heuristic for Improving Network Partitions , 1982, 19th Design Automation Conference.