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

Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing

Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM

Evaluation of a Flow-Based Hypergraph Bipartitioning Algorithm

In this paper, we propose HyperFlowCutter, an algorithm for balanced hypergraph bipartitioning that 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 new algorithm which obtains an initial partition with the fast multilevel hypergraph partitioner PaToH and then improves it using HyperFlowCutter as a refinement algorithm. ReBaHFC is able to significantly improve the solution quality of PaToH at little additional running time. The solution quality is only marginally worse than that of the best-performing hypergraph partitioners KaHyPar and hMETIS, while being one order of magnitude faster. Thus ReBaHFC offers a new time-quality trade-off in the current spectrum of hypergraph partitioners. For the special case of perfectly balanced bipartitioning, only the much slower plain HyperFlowCutter yields slightly better solutions than ReBaHFC, while only PaToH is faster than ReBaHFC

New Algorithmic Results in Clustering and Partitioning

Clustering and partitioning tasks have found widespread applications across computing. In machine learning, clustering represents the quintessential unsupervised learning task: grouping similar data points to discover structure in data. In operations research and combinatorial optimization, one is often interested in finding bottlenecks in a network, to identify possible weakness and points of failure. In this work, we discuss recent progress in better understanding computational aspects of clustering and partitioning. Our primary goal is establishing formal mathematical guarantees on the performance of clustering algorithms, as well as proving impossibility results to determine the inherent hardness of the problems we consider. In the first part of the thesis, we discuss graph partitioning tasks, focusing on the theory behind finding small vertex separators: few vertices which, when removed, disconnect the graph into large pieces. We design approximation algorithms for this problem, based on rounding natural convex relaxations. We also outline a recently uncovered connection between this problem and the fastest mixing random walk process on a graph with a target stationary distribution. In the second part of this work we discuss some algorithmic results in partitioning hypergraphs. We introduce a new, expressive class of hypergraph cut functions. We then design approximation algorithms for hypergraph generalizations of the minimum conductance cut problem by leveraging and extending techniques from spectral graph theory to the hypergraph regime. We prove our results for all the cut functions in our newly-defined class. In the process, we also improve on a popular primal-dual algorithmic framework for graph partitioning algorithms. Finally, we address the problem of learning partitions in an interactive way, by querying a same-cluster oracle, which determines whether two points belong to the same cluster. In this context we develop and analyze novel error-resistant algorithms, and provide complementary lower bounds, showing that our algorithms achieve optimal query complexity. To this end, we develop a new analytic framework based on modeling this task as a Rényi-Ulam liar game