Engineering Multilevel Graph Partitioning Algorithms

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.

💡 Research Summary

The paper introduces a novel multilevel graph partitioning framework that combines advanced local improvement techniques with global search strategies borrowed from the multigrid community. Traditional multilevel partitioners operate in three phases: coarsening, initial partitioning on a reduced graph, and refinement (often using FM or KL heuristics). While fast, these methods frequently stall at local optima, especially on large, irregular graphs.

To overcome this limitation, the authors propose two key innovations. First, they embed a max‑flow/min‑cut based local improvement step into the refinement phase. After an initial partition is obtained, the algorithm extracts a subgraph surrounding the current cut boundary. By designating the two sides of the partition as source and sink, a max‑flow computation (using push‑relabel, Dinic, or similar algorithms) yields a minimum cut that can be substantially larger than a single‑node move. The resulting cut is then used to relocate a group of vertices, effectively performing a more powerful, structure‑aware refinement. Because the flow is computed only on a localized subgraph, the additional computational overhead remains modest.

Second, the paper adapts multigrid concepts to the global search stage. In a multigrid hierarchy, a problem is solved on progressively coarser representations, capturing global structure before being interpolated back to finer levels. Here, the coarsening hierarchy generated during the multilevel process is reused: on each coarse level the authors apply the same flow‑based local improvement, producing a high‑quality partition at that scale. When moving back to the original fine graph, the coarse solution is not simply projected; instead, it is refined again with the flow‑based step, ensuring that global insights survive while fine‑grained adjustments are still possible. This synergy between coarse‑level global optimization and fine‑level local refinement yields partitions that are both balanced and have low edge cuts.

The experimental evaluation focuses on the well‑known Walshaw benchmark suite, which contains a variety of circuit and mesh graphs. The authors test three imbalance tolerances: 1 %, 3 %, and 5 %. Their algorithm improves upon the best known cuts in 317 instances across these settings and exactly reproduces the best known cut in an additional 118 cases. Compared with leading partitioners such as Metis, Scotch, and KaHIP, the proposed method delivers comparable or slightly faster runtimes while achieving superior cut quality.

Key contributions of the work are: (1) the integration of max‑flow/min‑cut based moves into the FM refinement loop, providing a more potent local search; (2) the novel use of multigrid‑style coarse‑to‑fine transitions for global exploration; (3) a thorough empirical demonstration that the combined approach consistently outperforms state‑of‑the‑art tools on standard benchmarks. The authors argue that the method is broadly applicable to domains requiring high‑quality partitions, including parallel task scheduling, VLSI placement, and community detection in social networks. Future directions suggested include GPU‑accelerated flow computations to further reduce runtime and extensions to dynamic graphs where partitions must be updated incrementally.