[Paper Review] Engineering Multilevel Graph Partitioning Algorithms

Peter Sanders, Christian Schulz|arXiv (Cornell University)|Nov 30, 2010

Advanced Graph Theory Research4 citations

TL;DR

This paper presents KaFFPa, a multilevel graph partitioning algorithm that combines max-flow min-cut based local improvement with global search strategies inspired by multigrid solvers. It achieves state-of-the-art partitioning quality, improving on 317 of 295 benchmark instances in Walshaw's archive and reproducing the best-known results in 118 cases, while also being faster than prior systems like kMetis and DiBaP.

ABSTRACT

We present a multi-level graph partitioning algorithm using novel local improvement algorithms and global search strategies transferred from the multi-grid community. Local improvement algorithms are based max-flow min-cut computations and more localized FM searches. By combining these techniques, we obtain an algorithm that is fast on the one hand and on the other hand is able to improve the best known partitioning results for many inputs. For example, in Walshaw's well known benchmark tables we achieve 317 improvements for the tables 1%, 3% and 5% imbalance. Moreover, in 118 additional cases we have been able to reproduce the best cut in this benchmark.

Motivation & Objective

• To develop a high-quality, fast multilevel graph partitioning algorithm that improves upon existing heuristic methods.
• To investigate how techniques from multigrid linear solvers—particularly global search strategies—can enhance graph partitioning.
• To design novel local improvement algorithms based on max-flow min-cut and localized FM searches for better partitioning quality.
• To evaluate the performance of the proposed algorithm on standard benchmarks, focusing on cut size and runtime.
• To achieve both superior solution quality and high sequential speed for large-scale graphs.

Proposed method

• The algorithm uses a multilevel approach: coarsening the graph via edge matching, initial partitioning on the coarsest level, and uncoarsening with local refinement.
• Local improvement is driven by max-flow min-cut computations to identify high-quality block boundaries.
• A more localized variant of the FM (Fiduccia-Mattheyses) algorithm is used to refine partitions efficiently.
• Global search strategies are adapted from multigrid solvers to explore the solution space more effectively.
• The system is implemented in C++ as KaFFPa (Karlsruhe Fast Flow Partitioner), with configurable modes for quality or speed.
• The algorithm is tuned using extensive experiments on real-world and synthetic graphs, including large-scale sequences.

Experimental results

Research questions

• RQ1 Can max-flow min-cut based local search significantly improve graph partitioning quality compared to traditional FM methods?
• RQ2 How do multigrid-inspired global search strategies enhance the exploration of the partitioning solution space?
• RQ3 To what extent can simple yet effective local improvement techniques outperform complex heuristics in multilevel partitioning?
• RQ4 How does the performance of the proposed algorithm scale with increasing graph size and number of blocks?
• RQ5 Can the algorithm achieve both high solution quality and fast runtime across diverse benchmark instances?

Key findings

• KaFFPa achieved 317 improvements over the best-known results in Walshaw’s benchmark tables for 1%, 3%, and 5% imbalance, with 99, 108, and 110 improvements respectively.
• In 118 out of 295 remaining cases, KaFFPa reproduced the best-known cut size, demonstrating strong consistency.
• On the largest graphs (e.g., european road network), KaFFPa Fast outperformed kMetis by up to 3× in cut size and 70% in runtime for k=64.
• For random geometric graphs, KaFFPa Fast reduced runtime by up to 24% and cut size by 6.5% compared to kMetis on the largest instances.
• The strongest configuration (KaFFPa Strong) achieved up to 47% better cut quality than kMetis on rgg graphs as size increased.
• For large graphs (≥78,000 nodes), KaFFPa achieved lower or equal cuts in 92% of cases, with the largest absolute improvement being 10% on add32 for k=4.