Browsing by Subject "Graph partitioning by vertex separator"

Now showing 1 - 4 of 4

Open Access
Partitioning hypergraphs in scientific computing applications through vertex separators on graphs
(Society for Industrial and Applied Mathematics, 2012) Kayaaslan, E.; Pinar, A.; Çatalyürek, U.; Aykanat, Cevdet
The modeling flexibility provided by hypergraphs has drawn a lot of interest from the combinatorial scientific community, leading to novel models and algorithms, their applications, and development of associated tools. Hypergraphs are now a standard tool in combinatorial scientific computing. The modeling flexibility of hypergraphs, however, comes at a cost: algorithms on hypergraphs are inherently more complicated than those on graphs, which sometimes translates to nontrivial increases in processing times. Neither the modeling flexibility of hypergraphs nor the runtime efficiency of graph algorithms can be overlooked. Therefore, the new research thrust should be how to cleverly trade off between the two. This work addresses one method for this trade-off by solving the hypergraph partitioning problem by finding vertex separators on graphs. Specifically, we investigate how to solve the hypergraph partitioning problem by seeking a vertex separator on its net intersection graph (NIG), where each net of the hypergraph is represented by a vertex, and two vertices share an edge if their nets have a common vertex. We propose a vertex-weighting scheme to attain good node-balanced hypergraphs, since the NIG model cannot preserve node-balancing information. Vertex-removal and vertex-splitting techniques are described to optimize cut-net and connectivity metrics, respectively, under the recursive bipartitioning paradigm. We also developed implementations of our proposed hypergraph partitioning formulations by adopting and modifying a state-of-the-art graph partitioning by vertex separator tool onmetis. Experiments conducted on a large collection of sparse matrices demonstrate the effectiveness of our proposed techniques. (c) 2012 Society for Industrial and Applied Mathematics.
Open Access
Permuting sparse rectangular matrices into block-diagonal form
(SIAM, 2004) Aykanat, Cevdet; Pınar, A.; Çatalyürek Ü. V.
We investigate the problem of permuting a sparse rectangular matrix into block-diagonal form. Block-diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization, and QR factorization. To represent the nonzero structure of a matrix, we propose bipartite graph and hypergraph models that reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning, respectively. Our experiments on a wide range of matrices, using the state-of-the-art graph and hypergraph partitioning tools MeTiS and PaToH, revealed that the proposed methods yield very effective solutions both in terms of solution quality and runtime.
Open Access
A recursive bipartitioning algorithm for permuting sparse square matrices into block diagonal form with overlap
(Society for Industrial and Applied Mathematics, 2013) Acer, S.; Kayaaslan, E.; Aykanat, Cevdet
We investigate the problem of symmetrically permuting a square sparse matrix into a block diagonal form with overlap. This permutation problem arises in the parallelization of an explicit formulation of the multiplicative Schwarz preconditioner and a more recent block overlapping banded linear solver as well as its application to general sparse linear systems. In order to formulate this permutation problem as a graph theoretical problem, we define a constrained version of the multiway graph partitioning by vertex separator (GPVS) problem, which is referred to as the ordered GPVS (oGPVS) problem. However, existing graph partitioning tools are unable to solve the oGPVS problem. So, we also show how the recursive bipartitioning framework can be utilized for solving the oGPVS problem. For this purpose, we propose a left-to-right bipartitioning approach together with a novel vertex fixation scheme so that existing 2-way GPVS tools that support fixed vertices can be effectively and efficiently utilized in the recursive bipartitioning framework. Experimental results on a wide range of matrices confirm the validity of the proposed approach. © 2013 Society for Industrial and Applied Mathematics.
Open Access
A recursive graph bipartitioning algorithm by vertex separators with fixed vertices for permuting sparse matrices into block diagonal form with overlap
(2011) Acer, Seher
Solving sparse system of linear equations Ax=b using preconditioners can be effi- ciently parallelized using graph partitioning tools. In this thesis, we investigate the problem of permuting a sparse matrix into a block diagonal form with overlap which is to be used in the parallelization of the multiplicative schwarz preconditioner. A matrix is said to be in block diagonal form with overlap if the diagonal blocks may overlap. In order to formulate this permutation problem as a graph-theoretical problem, we introduce a restricted version of the graph partitioning by vertex separator problem (GPVS), where the objective is to find a vertex partition whose parts are only connected by a vertex separator. The modified problem, we refer as ordered GPVS problem (oGPVS), is restricted such that the parts should exhibit an ordered form where the consecutive parts can only be connected by a separator. The existing graph partitioning tools are unable to solve the oGPVS problem. Thus, we present a recursive graph bipartitioning algorithm by vertex separators together with a novel vertex fixation scheme so that a GPVS tool supporting fixed vertices can effectively and efficiently be utilized. We also theoretically verified the correctness of the proposed approach devising a necessary and sufficient condition to the feasibility of a oGPVS solution. Experimental results on a wide range of matrices confirm the validity of the proposed approach.