[Paper Review] Hypergraph Partitioning through Vertex Separators on Graphs

Enver Kayaaslan, Ali Pınar+2 more|arXiv (Cornell University)|Mar 1, 2011

VLSI and FPGA Design Techniques55 references3 citations

TL;DR

This paper proposes a novel hypergraph partitioning approach by modeling the problem as vertex separator detection on the net intersection graph (NIG) of a hypergraph. By leveraging graph partitioning by vertex separator (GPVS) techniques and introducing vertex-weighting and vertex-splitting strategies, the method achieves up to 4.17x speedup over PaToH while maintaining comparable cutsize quality and acceptable load balancing, especially for large partition counts.

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 translate 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 NIG model cannot preserve node balancing information. Vertex-removal and vertex-splitting techniques are described to optimize cutnet and connectivity metrics, respectively, under the recursive bipartitioning paradigm. We also developed an implementation for our GPVS-based HP formulations by adopting and modifying a state-of-the-art GPVS tool onmetis. Experiments conducted on a large collection of sparse matrices confirmed the validity of our proposed techniques.

Motivation & Objective

• To address the trade-off between hypergraph modeling flexibility and graph algorithm runtime efficiency.
• To develop a method that enables efficient hypergraph partitioning using existing graph partitioning tools.
• To preserve cutsize quality while significantly reducing partitioning time compared to state-of-the-art hypergraph partitioners.
• To provide a practical solution for applications where computational balance is not strictly defined or where preprocessing overhead is critical.

Proposed method

• The hypergraph is transformed into its net intersection graph (NIG), where each net is a vertex and edges represent shared vertices between nets.
• A vertex-weighting scheme is introduced to approximate node balancing, since NIGs inherently lose vertex weight information.
• Vertex-removal and vertex-splitting techniques are applied during recursive bipartitioning to optimize cutnet and connectivity metrics.
• The approach integrates with the state-of-the-art GPVS tool 'metis' by modifying it into 'onmetisHP' to support hypergraph partitioning via NIG-based vertex separators.
• The recursive bipartitioning paradigm is used to build K-way partitions, with free-node-to-part assignments performed after each 2-way GPVS step to ensure accurate splitting.
• The method amortizes NIG construction overhead, leading to increasing speedup with larger K values.

Experimental results

Research questions

• RQ1 Can hypergraph partitioning be effectively solved by reducing it to vertex separator problems on the net intersection graph?
• RQ2 How can node balancing be approximated in the NIG model, which inherently loses vertex weight information?
• RQ3 To what extent can GPVS-based methods match or exceed the cutsize quality of state-of-the-art hypergraph partitioners like PaToH?
• RQ4 How does the performance of the proposed method scale with increasing partition count K, especially in terms of speedup and load imbalance?

Key findings

• The proposed method achieved up to 4.17x speedup over PaToH on the PD matrix collection, with the highest speedup observed at K=128.
• Cutsize quality was preserved on average and improved in many cases compared to PaToH, with average cutsize ratios of 1.03 to 1.63 across different K values.
• Load imbalance (LI) was worse than PaToH’s for larger K, but remained acceptable, with %LI values ranging from 0.1% to 16.5% depending on K and metric.
• Speedup increased with K due to amortization of NIG construction overhead, indicating better scalability for large-scale partitioning.
• The method demonstrated that cutsize and runtime can be traded off against balance, enabling effective partitioning even for high K with acceptable balance.
• Empirical results confirm that the GPVS-based hypergraph partitioning formulation is competitive and highly efficient, especially in scenarios where precise load balancing is less critical.