Weighted aggregation for multi-level graph partitioning

Graph partitioning is a well-known optimization problem of great interest in theoretical and applied studies. Since the 1990s, many multilevel schemes have been introduced as a practical tool to solve this problem. A multilevel algorithm may be viewed as a process of graph topology learning at different scales in order to generate a better approximation for any approximation method incorporated at the uncoarsening stage in the framework. In this work we compare two multilevel frameworks based on the geometric and the algebraic multigrid schemes for the partitioning problem

Recent Advances in Graph Partitioning

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

Multilevel spectral clustering : graph partitions and image segmentation

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008.Includes bibliographical references (p. 145-146).While the spectral graph partitioning method gives high quality segmentation, segmenting large graphs by the spectral method is computationally expensive. Numerous multilevel graph partitioning algorithms are proposed to reduce the segmentation time for the spectral partition of large graphs. However, the greedy local refinement used in these multilevel schemes has the tendency of trapping the partition in poor local minima. In this thesis, I develop a multilevel graph partitioning algorithm that incorporates the inverse powering method with greedy local refinement. The combination of the inverse powering method with greedy local refinement ensures that the partition quality of the multilevel method is as good as, if not better than, segmenting the large graph by the spectral method. In addition, I present a scheme to construct the adjacency matrix, W and degree matrix, D for the coarse graphs. The proposed multilevel graph partitioning algorithm is able to bisect a graph (k = 2) with significantly shorter time than segmenting the original graph without the multilevel implementation, and at the same time achieving the same normalized cut (Ncut) value. The starting eigenvector, obtained by solving a generalized eigenvalue problem on the coarsest graph, is close to the Fiedler vector of the original graph. Hence, the inverse iteration needs only a few iterations to converge the starting vector. In the k-way multilevel graph partition, the larger the graph, the greater the reduction in the time needed for segmenting the graph. For the multilevel image segmentation, the multilevel scheme is able to give better segmentation than segmenting the original image. The multilevel scheme has higher success of preserving the salient part of an object.(cont.) In this work, I also show that the Ncut value is not the ultimate yardstick for the segmentation quality of an image. Finding a partition that has lower Ncut value does not necessary means better segmentation quality. Segmenting large images by the multilevel method offers both speed and quality.by Tian Fook Kong.S.M

Partitioning Complex Networks via Size-constrained Clustering

The most commonly used method to tackle the graph partitioning problem in practice is the multilevel approach. During a coarsening phase, a multilevel graph partitioning algorithm reduces the graph size by iteratively contracting nodes and edges until the graph is small enough to be partitioned by some other algorithm. A partition of the input graph is then constructed by successively transferring the solution to the next finer graph and applying a local search algorithm to improve the current solution. In this paper, we describe a novel approach to partition graphs effectively especially if the networks have a highly irregular structure. More precisely, our algorithm provides graph coarsening by iteratively contracting size-constrained clusterings that are computed using a label propagation algorithm. The same algorithm that provides the size-constrained clusterings can also be used during uncoarsening as a fast and simple local search algorithm. Depending on the algorithm's configuration, we are able to compute partitions of very high quality outperforming all competitors, or partitions that are comparable to the best competitor in terms of quality, hMetis, while being nearly an order of magnitude faster on average. The fastest configuration partitions the largest graph available to us with 3.3 billion edges using a single machine in about ten minutes while cutting less than half of the edges than the fastest competitor, kMetis

Tree-based Coarsening and Partitioning of Complex Networks

Many applications produce massive complex networks whose analysis would benefit from parallel processing. Parallel algorithms, in turn, often require a suitable network partition. For solving optimization tasks such as graph partitioning on large networks, multilevel methods are preferred in practice. Yet, complex networks pose challenges to established multilevel algorithms, in particular to their coarsening phase. One way to specify a (recursive) coarsening of a graph is to rate its edges and then contract the edges as prioritized by the rating. In this paper we (i) define weights for the edges of a network that express the edges' importance for connectivity, (ii) compute a minimum weight spanning tree $T^m$ with respect to these weights, and (iii) rate the network edges based on the conductance values of $T^m$'s fundamental cuts. To this end, we also (iv) develop the first optimal linear-time algorithm to compute the conductance values of \emph{all} fundamental cuts of a given spanning tree. We integrate the new edge rating into a leading multilevel graph partitioner and equip the latter with a new greedy postprocessing for optimizing the maximum communication volume (MCV). Experiments on bipartitioning frequently used benchmark networks show that the postprocessing already reduces MCV by 11.3%. Our new edge rating further reduces MCV by 10.3% compared to the previously best rating with the postprocessing in place for both ratings. In total, with a modest increase in running time, our new approach reduces the MCV of complex network partitions by 20.4%

Parallel Graph Partitioning for Complex Networks

Processing large complex networks like social networks or web graphs has recently attracted considerable interest. In order to do this in parallel, we need to partition them into pieces of about equal size. Unfortunately, previous parallel graph partitioners originally developed for more regular mesh-like networks do not work well for these networks. This paper addresses this problem by parallelizing and adapting the label propagation technique originally developed for graph clustering. By introducing size constraints, label propagation becomes applicable for both the coarsening and the refinement phase of multilevel graph partitioning. We obtain very high quality by applying a highly parallel evolutionary algorithm to the coarsened graph. The resulting system is both more scalable and achieves higher quality than state-of-the-art systems like ParMetis or PT-Scotch. For large complex networks the performance differences are very big. For example, our algorithm can partition a web graph with 3.3 billion edges in less than sixteen seconds using 512 cores of a high performance cluster while producing a high quality partition -- none of the competing systems can handle this graph on our system.Comment: Review article. Parallelization of our previous approach arXiv:1402.328