Source-linked AI summary
Engineering Multilevel Graph Partitioning Algorithms
Peter Sanders, Christian Schulz
TL;DR
Large-graph partitioning requires balanced blocks with few inter-block edges, but multilevel methods leave choices about local improvement and global search. The paper introduces KaFFPa with flow-based and localized FM refinement plus multigrid-inspired search strategies. It reports best-known partitions for many graphs, including 317 Walshaw benchmark improvements and 118 reproduced best cuts.
Problem
The paper addresses balanced partitioning of large graphs while minimizing edges between blocks, with multilevel methods offering several aspects to reassess for quality and speed.
Method
KaFFPa combines max-flow min-cut and localized FM-based local improvement methods with global search strategies transferred from multigrid methods.
Results
KaFFPa computed 317 partitions better than previous Walshaw benchmark results and reproduced equally sized cuts in 118 additional cases.
Takeaways & Limitations
KaFFPa computes best-known partitions for many graphs when some imbalance is allowed and can be configured for high quality or sequential speed.
Takeaways & Limitations
The authors did not handle ϵ = 0, which may require different local search strategies, and describe the implementation as a research prototype.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper studies balanced graph partitioning for minimizing inter-block edges and revisits multilevel methods to improve solution quality and sequential speed. KaFFPa combines new local refinement and global search strategies, with configurations targeting either high quality or speed.
- Problem: Graph partitioning divides computation and communication graphs into approximately equal blocks while minimizing edges crossing between blocks.The formulation limits maximum block size to (1 + ϵ) times the average block size.
- Multilevel approach: Multilevel graph partitioning recursively contracts a graph, partitions the smallest graph, then refines the partition while undoing contractions.Each coarser level is intended to preserve the input graph’s basic structure.
- Motivation: The authors reassess multilevel partitioning because some aspects were poorly understood and simple methods unexpectedly improved partition quality.Their stated focus is solution quality and sequential speed for large graphs, while deferring parallelization.
- Contribution: KaFFPa can be configured either to achieve best-known partitions for many benchmark instances or to run fastest on large graphs while improving quality over the previous fastest system.The paper reports these as alternative quality- and speed-oriented configurations.
- Evaluation: KaFFPa implements the techniques in C++ and experiments indicate that it scales to large networks while computing very high-quality partitions.The system includes novel local improvement and global search methods described in later sections.
2 Preliminaries
The preliminaries define the balanced cut objective and describe KaFFPa’s multilevel pipeline: contraction, initial partitioning, uncoarsening, and local refinement. Refinement combines quotient-graph pairwise methods with two-way and k-way local searches.
- Partitioning problem: A feasible partition consists of k disjoint blocks whose weights satisfy c(V_i) ≤ (1 + ϵ)c(V)/k + max_v∈V c(v), while minimizing total cut weight.The additive term accounts for indivisible nodes, especially the heaviest node.
- Contraction: A matching is a set of edges sharing no endpoints, and contracting a matched edge replaces its two endpoints with one node.The new node aggregates endpoint weights and combines parallel-edge weights.
- Multilevel pipeline: The multilevel pipeline contracts matchings until the graph is small, partitions the reduced graph, then uncontracts matchings and refines the projected partition.Contraction aims to preserve global structure by making nodes represent densely connected subgraphs.
- Contraction and initialization: KaFFPa uses edge ratings, the Global Path Algorithm for near-linear-time matching, and Scotch for initial partitioning.GPA optimally solves selected paths and even-length cycles using dynamic programming and was empirically stronger than several alternative matchings.
- Refinement: During refinement, quotient-graph style improvement applies two-way methods to block pairs sharing a boundary, enabling flow-based techniques between those blocks.The quotient graph represents blocks as vertices and inter-block connectivity as edges.
- Local search: The k-way local search uses one priority queue, moves each node at most once per search, rejects moves causing imbalance, and rolls back to the lowest feasible cut.Iterations continue until no improvement is found or a maximum iteration count is reached.
3 Related Work
Related work situates KaFFPa among multilevel graph partitioners, flow-based cut-improvement methods, and non-multilevel approaches. Prior systems differ in their contraction granularity, refinement mechanism, or target cut objective.
- Multilevel methods: Most general-purpose methods for high-quality partitioning of large real-world graphs use the multilevel principle based on contraction and local search.Examples include Chaco, Jostle, Metis, Party, and Scotch.
- Multilevel partitioners: KaSPar uses single-edge contraction between levels, while KaPPa is a matching-based multilevel algorithm designed for scalable parallel execution.KaPPa’s local search considers independent block pairs at a time.
- Alternative refinement: DiBaP uses diffusion-based local improvement and also produces partitions of very high quality.Its refinement mechanism differs from the local-search and flow-based approaches emphasized in this paper.
- Flow-based methods: MQI and Improve use max-flow methods to improve quotient-style objectives such as expansion or conductance, but their approach is currently feasible only for k = 2.Improve consistently matches or exceeds MQI in the cited comparison.
- Non-multilevel methods: PUNCH is not multilevel but constructs a coarse graph from natural cuts and performs well on road-network instances.Natural cuts are sparse cuts near denser regions, identified through minimum cuts between selected regions.
- Iterated multilevel methods: Iterated multilevel algorithms reuse a partition during later coarsening and uncoarsening, preventing inter-block edges from being matched or contracted.The approach is intended to increase quality when refinement does not worsen the initial partition.
4 Local Improvement
The paper introduces two local improvement approaches for multilevel graph partitioning: flow-based refinement between block pairs and localized k-way search. The flow method constructs corridor-based max-flow problems to obtain balanced improved cuts, while a residual-graph heuristic searches for better-balanced minimum cuts.
- Overview: Two novel methods are introduced: pairwise max-flow min-cut refinement and multi-try FM, a more localized k-way search.The pairwise method improves a given 2-partition; multi-try FM repeatedly starts from a small random boundary neighborhood.
- Overview: After uncontracting a matching, local improvement reduces the projected cut while preserving the balance constraint.The refinement stage applies local methods after each uncoarsening step.
- Flow-based refinement: The flow-based method builds a corridor-induced graph G′ whose minimum cuts induce feasible cuts in the original graph G.The corridor is selected using breadth-first searches, then augmented with directed infinite-weight source and sink edges to enforce feasibility.
- Flow-based refinement: Adaptive flow iterations enlarge or shrink the corridor to search for balanced cuts beyond the initial local neighborhood.An improved feasible cut can trigger another iteration, while the corridor size is adjusted according to whether the resulting cut satisfies the original balance constraint.
- Most balanced minimum cuts: A maximum flow supports enumeration of minimum cuts through closed vertex sets in a reduced residual graph.Strongly connected components are contracted, forced source- and sink-side components are removed, and randomized topological sweeps generate minimum cuts with different balances.
- Most balanced minimum cuts: Finding the minimum cut with the best balance is NP-hard, and larger corridors can produce minimum cuts that are infeasible for the original partitioning problem.The heuristic therefore returns the best-balanced feasible cut found during repeated searches.
5 Global Search
The paper extends iterated multilevel graph partitioning beyond V-cycles with W- and F-cycle global search strategies. Runtime analysis shows that cycle costs depend on shrink factor and the level split parameter, with F-cycles potentially much cheaper than W-cycles under optimistic assumptions.
- V-cycles: Iterated multilevel algorithms reuse an available partition during coarsening and uncoarsening, producing V-cycles that can improve quality across randomized iterations.Edges between blocks are not matched or contracted, allowing the partition to initialize the coarsest graph; refinement is required not to worsen it.
- W- and F-cycles: W-cycles and F-cycles transfer multigrid-style global search strategies to graph partitioning by performing additional trials on coarser graphs.The figure contrasts a single V-cycle with W- and F-cycle structures, while the analysis characterizes their runtime recursively.
- Runtime analysis: Theorem 1 analyzes V-, W-, and F-cycle runtimes for linear coarsening and refinement time with constant shrink factor a ∈[1/2, 1).The W- and F-cycle bounds depend on the level split parameter d and the relationship between 2ad and 1.
- Runtime analysis: The W-cycle runtime includes d coarsening and refinement steps plus two trials on the created coarse graph.This recurrence is analyzed separately according to whether 2ad is less than, equal to, or greater than 1.
- Runtime analysis: The runtime analysis is completed using the total number of levels l and divide-and-conquer reasoning for the W-cycle and F-cycle recurrences.The proof explicitly identifies l as the total number of levels.
- Runtime trade-offs: With a = 1/2 and d = 1, an F-cycle is twice as expensive as a single V-cycle, whereas a W-cycle has a factor log n larger asymptotic execution time.Because practical shrink factors are usually worse than 1/2, W-cycle costs can grow further; the level split parameter d is therefore crucial.
6 Experiments
Experiments evaluate KaFFPa’s flow-based local improvement, global search, component contributions, and comparisons with established partitioners. The results show configurable quality–runtime trade-offs, strong benchmark performance, and advantages on large graphs.
- Insights about Flows: KaFFPa’s flow-plus-local-search configurations produce up to 6.14% lower cuts, or 7.21% with the most balancing heuristic, than basic two-way FM.The strongest effectiveness configuration yields 4.73% lower cuts than basic FM, despite increased runtime.
- Insights about Global Search Strategies: More sophisticated global search strategies reduce cuts but increase runtime; repeated 2 W-cycles and 2 F-cycles achieve 1.5% lower best cuts than repeated normal V-cycles.Reusing partitions and active-block scheduling contribute to the increased effectiveness and later-cycle speedups.
- Removal / Knockout Tests: Removing all new components—global k-way search, multitry search, F-Cycles, and Flow—produces 9.3% larger cuts than KaFFPa Strong.Flow and multitry search produce the largest differences in the removal and knockout tests.
- Comparison with other Partitioners: KaFFPa Strong produces lower cuts than kMetis, Scotch, DiBaP, KaPPa, and KaSPar by approximately 33%, 20%, 11%, 12%, and 3%, respectively.KaFFPa is better than KaSPar’s best cut in 57 of 66 cases.
- Comparison with other Partitioners: On the largest available graphs, KaFFPa Fast outperforms kMetis in both quality and runtime; on the European road network, kMetis runs 44% longer and produces up to 3× larger cuts.For increasing random-geometric graph sizes, KaFFPa Fast reaches 2.5% smaller cuts while kMetis becomes 70% slower on the largest graph.
- The Walshaw Benchmark: KaFFPa improves 317 Walshaw benchmark partitions and reproduces equally sized cuts in 118 additional cases across 1%, 3%, and 5% imbalance.The improvements comprise 99, 108, and 110 cases for 1%, 3%, and 5% imbalance, respectively.
7 Conclusions and Future Work
KaFFPa achieves best-known partitions for many graphs under certain imbalance allowances through new local improvement methods and global search strategies. The authors identify untested zero-imbalance handling, further integration opportunities, and prototype status as boundaries and future work.
- KaFFPa currently computes the best-known partitions for many graphs when a certain imbalance is allowed.
- Its success combines max-flow min-cut-based local improvement, localized local searches, and global search strategies transferred from multigrid linear solvers.
- The authors did not try the zero-imbalance case, which may require different local search strategies.
- Future work includes parallel graph partitioning, additional initial partitioning and refinement algorithms, and integration with evolutionary search.
- The current KaFFPa implementation is a research prototype rather than a widely usable tool.