Partitioning Hypergraphs in Scientific Computing Applications through Vertex Separators on Graphs

Abstract

Abstract. 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.

A survey of direct methods for sparse linear systems

“…However, until recently methods to order hypergraphs were limited. A net intersection graph hypergraph model, where each net in the original hypergraph is represented by a vertex and each vertex of the original hypergraph is replaced by a hyperedge representing a clique of all the neighbours of the original vertex, established the relationship between vertex separators and hypergraph partitioning (Kayaaslan, Pinar, Çatalyürek and Aykanat 2012). Later, methods based on hypergraph partitioning were used to compute vertex separators (Çatalyürek, Aykanat and Kayaaslan 2011).…”

An exact algorithm for sparse matrix bipartitioning

“…Kayaaslan et al [27] present a heuristic method for hypergraph partitioning based on graph partitioning by vertex separator (GPVS), a procedure which for p = 2 partitions the vertices of a graph into two unconnected sets and a separator set. This is similar to our approach of partitioning the rows and columns into three sets, although we use it to obtain an optimal partitioning instead of a heuristic one.…”

SciDAC Institute: Combinatorial Scientific Computing and Petascale Simulations (CSCAPES) (Final Report)

“…We developed a new technique for solving hypergraph partitioning problems by using GPVS [18]. GPVS formulation cannot preserve exact load--balancing information but allows us to trade--off partitioning quality for execution time.…”