Advanced Multilevel Node Separator Algorithms

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 highly localized local search and iterated multilevel cycles to improve solution quality even further. Experiments indicate that flow-based local search algorithms on its own in a multilevel framework are already highly competitive in terms of separator quality. Adding additional local search algorithms further improves solution quality. Our strongest configuration almost always outperforms competing systems while on average computing 10% and 62% smaller separators than Metis and Scotch, respectively

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

Multilevel Combinatorial Optimization Across Quantum Architectures

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 multilevel frameworks for combinatorial optimization as a promising general paradigm for designing hybrid quantum-classical algorithms. In order to demonstrate this approach, we apply this method to two well-known combinatorial optimization problems, namely, the Graph Partitioning Problem, and the Community Detection Problem. We develop hybrid multilevel solvers with quantum local search on D-Wave's quantum annealer and IBM's gate-model based quantum processor. We carry out experiments on graphs that are orders of magnitudes larger than the current quantum hardware size, and we observe results comparable to state-of-the-art solvers in terms of quality of the solution

Computational Optimization Techniques for Graph Partitioning

Partitioning graphs into two or more subgraphs is a fundamental operation in computer science, with applications in large-scale graph analytics, distributed and parallel data processing, and fill-reducing orderings in sparse matrix algorithms. Computing balanced and minimally connected subgraphs is a common pre-processing step in these areas, and must therefore be done quickly and efficiently. Since graph partitioning is NP-hard, heuristics must be used. These heuristics must balance the need to produce high quality partitions with that of providing practical performance. Traditional methods of partitioning graphs rely heavily on combinatorics, but recent developments in continuous optimization formulations have led to the development of hybrid methods that combine the best of both approaches. This work describes numerical optimization formulations for two classes of graph partitioning problems, edge cuts and vertex separators. Optimization-based formulations for each of these problems are described, and hybrid algorithms combining these optimization-based approaches with traditional combinatoric methods are presented. Efficient implementations and computational results for these algorithms are presented in a C++ graph partitioning library competitive with the state of the art. Additionally, an optimization-based approach to hypergraph partitioning is proposed

Fast Machine Learning Algorithms for Massive Datasets with Applications in the Biomedical Domain

The continuous increase in the size of datasets introduces computational challenges for machine learning algorithms. In this dissertation, we cover the machine learning algorithms and applications in large-scale data analysis in manufacturing and healthcare. We begin with introducing a multilevel framework to scale the support vector machine (SVM), a popular supervised learning algorithm with a few tunable hyperparameters and highly accurate prediction. The computational complexity of nonlinear SVM is prohibitive on large-scale datasets compared to the linear SVM, which is more scalable for massive datasets. The nonlinear SVM has shown to produce significantly higher classification quality on complex and highly imbalanced datasets. However, a higher classification quality requires a computationally expensive quadratic programming solver and extra kernel parameters for model selection. We introduce a generalized fast multilevel framework for regular, weighted, and instance weighted SVM that achieves similar or better classification quality compared to the state-of-the-art SVM libraries such as LIBSVM. Our framework improves the runtime more than two orders of magnitude for some of the well-known benchmark datasets. We cover multiple versions of our proposed framework and its implementation in detail. The framework is implemented using PETSc library which allows easy integration with scientific computing tasks. Next, we propose an adaptive multilevel learning framework for SVM to reduce the variance between prediction qualities across the levels, improve the overall prediction accuracy, and boost the runtime. We implement multi-threaded support to speed up the parameter fitting runtime that results in more than an order of magnitude speed-up. We design an early stopping criteria to reduce the extra computational cost when we achieve expected prediction quality. This approach provides significant speed-up, especially for massive datasets. Finally, we propose an efficient low dimensional feature extraction over massive knowledge networks. Knowledge networks are becoming more popular in the biomedical domain for knowledge representation. Each layer in knowledge networks can store the information from one or multiple sources of data. The relationships between concepts or between layers represent valuable information. The proposed feature engineering approach provides an efficient and highly accurate prediction of the relationship between biomedical concepts on massive datasets. Our proposed approach utilizes semantics and probabilities to reduce the potential search space for the exploration and learning of machine learning algorithms. The calculation of probabilities is highly scalable with the size of the knowledge network. The number of features is fixed and equivalent to the number of relationships or classes in the data. A comprehensive comparison of well-known classifiers such as random forest, SVM, and deep learning over various features extracted from the same dataset, provides an overview for performance and computational trade-offs. Our source code, documentation and parameters will be available at https://github.com/esadr/