High-Quality Multilevel Hypergraph Partitioning

KaHyPar is a multilevel hypergraph partitioning framework providing direct k-way and recursive bisection based partitioning algorithms that compute solutions of very high quality.

View the Project on GitHub SebastianSchlag/kahypar

KaHyPar - Karlsruhe Hypergraph Partitioning

What is a Hypergraph? What is Hypergraph Partitioning?

Hypergraphs are a generalization of graphs, where each (hyper)edge (also called net) can connect more than two vertices. The k-way hypergraph partitioning problem is the generalization of the well-known graph partitioning problem: partition the vertex set into k disjoint blocks of bounded size (at most 1 + ε times the average block size), while minimizing an objective function defined on the nets.

The two most prominent objective functions are the cut-net and the connectivity (or λ − 1) metrics. Cut-net is a straightforward generalization of the edge-cut objective in graph partitioning (i.e., minimizing the sum of the weights of those nets that connect more than one block). The connectivity metric additionally takes into account the actual number λ of blocks connected by a net. By summing the (λ − 1)-values of all nets, one accurately models the total communication volume of parallel sparse matrix-vector multiplication and once more gets a metric that reverts to edge-cut for plain graphs.  What is KaHyPar?

KaHyPar is a multilevel hypergraph partitioning framework for optimizing the cut- and the (λ − 1)-metric. It supports both recursive bisection and direct k-way partitioning. As a multilevel algorithm, it consist of three phases: In the coarsening phase, the hypergraph is coarsened to obtain a hierarchy of smaller hypergraphs. After applying an initial partitioning algorithm to the smallest hypergraph in the second phase, coarsening is undone and, at each level, a local search method is used to improve the partition induced by the coarser level. KaHyPar instantiates the multilevel approach in its most extreme version, removing only a single vertex in every level of the hierarchy. By using this very fine grained n-level approach combined with strong local search heuristics, it computes solutions of very high quality. Its algorithms and detailed experimental results are presented in several research publications.

• Hypergraph Partitioning with Variable Block Weights:
KaHyPar has support for variable block weights. If command line option --use-individual-part-weights=true is used, the partitioner tries to partition the hypergraph such that each block Vx has a weight of at most Bx, where Bx can be specified for each block individually using the command line parameter --part-weights= B1 B2 B3 ... Bk-1. Since the framework does not yet support perfectly balanced partitioning, upper bounds need to be slightly larger than the total weight of all vertices of the hypergraph. Note that this feature is still experimental.
• Hypergraph Partitioning with Fixed Vertices:

Hypergraph partitioning with fixed vertices is a variation of standard hypergraph partitioning. In this problem, there is an additional constraint on the block assignment of some vertices, i.e., some vertices are preassigned to specific blocks prior to partitioning with the condition that, after partitioning the remaining “free” vertices, the fixed vertices are still in the block that they were assigned to. The command line parameter --fixed / -f can be used to specify a fix file in hMetis fix file format. For a hypergraph with V vertices, the fix file consists of V lines - one for each vertex. The ith line either contains -1 to indicate that the vertex is free to move or <part id> to indicate that this vertex should be preassigned to block <part id>. Note that part ids start from 0.

KaHyPar currently supports three different contraction policies for partitioning with fixed vertices:

1. free_vertex_only allows all contractions in which the contraction partner is a free vertex, i.e., it allows contractions of vertex pairs where either both vertices are free, or one vertex is fixed and the other vertex is free.
2. fixed_vertex_allowed additionally allows contractions of two fixed vertices provided that both are preassigned to the same block. Based on preliminary experiments, this is currently the default policy.
3. equivalent_vertices only allows contractions of vertex pairs that consist of either two free vertices or two fixed vertices preassigned to the same block.
• Evolutionary Framework (KaHyPar-E):

KaHyPar-E enhances KaHyPar with an evolutionary framework as described in our GECCO’18 publication. Given a fairly large amount of running time, this memetic multilevel algorithm performs better than repeated executions of KaHyPar-MF/-CA, hMetis, and PaToH. The configuration /config/km1_direct_kway_gecco18.ini uses KaHyPar-CA to exploit the local solution space and was used in the GECCO’18 experiments. The command line parameter --time-limit=xxx can be used to set the maximum running time (in seconds). Parameter --partition-evolutionary=true enables evolutionary partitioning.

• Improve Existing Partitions:

KaHyPar uses direct k-way V-cycles to try to improve an existing partition specified via parameter --part-file=</path/to/file>. The maximum number of V-cycles can be controlled via parameter --vcycles=.

Experimental Results

We use the performance profiles to compare KaHyPar to other partitioning algorithms in terms of solution quality. For a set of algorithms and a benchmark set containing || instances, the performance ratio relates the cut computed by partitioner p for instance i to the smallest minimum cut of all algorithms, i.e.,

The performance profile of algorithm p is then given by the function

For connectivity optimization, the performance ratios are computed using the connectivity values instead of the cut values. The value of corresponds to the fraction of instances for which partitioner p computed the best solution, while is the probability that a performance ratio is within a factor of of the best possible ratio. Note that since performance profiles only allow to assess the performance of each algorithm relative to the best algorithm, the values cannot be used to rank algorithms (i.e., to determine which algorithm is the second best etc.).

In our experimental analysis, the performance profile plots are based on the best solutions (i.e., minimum connectivity/cut) each algorithm found for each instance. Furthermore, we choose parameters for all and such that a performance ratio if and only if algorithm p computed an infeasible solution for instance i, and if and only if the algorithm could not compute a solution for instance i within the given time limit. Since the performance ratios are heavily right-skewed, the performance profile plots are divided into three segments with different ranges for parameter to reflect various areas of interest. The first segment highlights small values (), while the second segment contains results for all instances that are up to a factor of worse than the best possible ratio. The last segment contains all remaining ratios, i.e., instances for which some algorithms performed considerably worse than the best algorithm, instances for which algorithms produced infeasible solutions, and instances which could not be partitioned within the given time limit. The last segment uses a log-scale on the x-axis.

In the figures, we compare KaHyPar with PaToH in quality (PaToH-Q) and default mode (PaToH-D), the k-way (hMetis-K) and the recursive bisection variant (hMetis-R) of hMetis 2.0 (p1), Zoltan using algebraic distance-based coarsening (Zoltan-AlgD) and the recently published HYPE algorithm.

Connectivity Optimization Cut-Net Optimization We provide additional resources for all KaHyPar-related publications:

KaHyPar-MF /
KaHyPar-R-MF
SEA’18 /
JEA’19
SEA Paper /
JEA Paper
TR Slides Experimental Results:
SEA / JEA
KaHyPar-E (EvoHGP) GECCO’18 Paper TR Slides Experimental Results
KaHyPar-CA SEA’17 Paper - Slides Experimental Results
KaHyPar-K ALENEX’17 Paper - Slides Experimental Results
KaHyPar-R ALENEX’16 Paper TR Slides Experimental Results

Requirements

The Karlsruhe Hypergraph Partitioning Framework requires:

• A 64-bit operating system. Linux, Mac OS X and Windows are currently supported.
• A modern, -ready compiler such as g++ version 5.2 or higher or clang version 3.2 or higher.
• The cmake build system.
• The Boost.Program_options library.

Building KaHyPar

1. Clone the repository including submodules:

git clone --depth=1 --recursive git@github.com:SebastianSchlag/kahypar.git

2. Create a build directory: mkdir build && cd build
3. Run cmake: cmake .. -DCMAKE_BUILD_TYPE=RELEASE
4. Run make: make

Testing and Profiling

Tests are automatically executed while project is built. Additionally a test target is provided. End-to-end integration tests can be started with: make integration_tests. Profiling can be enabled via cmake flag: -DENABLE_PROFILE=ON.

Running KaHyPar

The standalone program can be built via make KaHyPar. The binary will be located at: build/kahypar/application/.

KaHyPar has several configuration parameters. For a list of all possible parameters please run: ./KaHyPar --help. We use the hMetis format for the input hypergraph file as well as the partition output file.

Currently we provide nine different presets that correspond to the configurations used in the publications at ALENEX’16, ALENEX’17, SEA’17, SEA’18, GECCO’18, as well as our JEA journal paper (to appear).

To start KaHyPar-MF (using flow-based refinement) optimizing the (connectivity - 1) objective using direct k-way mode run:

./KaHyPar -h <path-to-hgr> -k <# blocks> -e <imbalance (e.g. 0.03)> -o km1 -m direct -p ../../../config/km1_kahypar_mf_jea19.ini

To start KaHyPar-MF (using flow-based refinement) optimizing the cut-net objective using direct k-way mode run:

./KaHyPar -h <path-to-hgr> -k <# blocks> -e <imbalance (e.g. 0.03)> -o cut -m direct -p ../../../config/cut_kahypar_mf_jea19.ini

To start EvoHGP/KaHyPar-E optimizing the (connectivity - 1) objective using direct k-way mode run

./KaHyPar -h <path-to-hgr> -k <# blocks> -e <imbalance (e.g. 0.03)> -o km1 -m direct -p ../../../config/km1_direct_kway_gecco18.ini

Note that the configuration km1_direct_kway_gecco18.ini is based on KaHyPar-CA. However, KaHyPar-E also works with flow-based local improvements. In our JEA publication the km1_kahypar_e_mf_jea19.ini configuration was used.

To start KaHyPar-CA (using community-aware coarsening) optimizing the (connectivity - 1) objective using direct k-way mode run:

./KaHyPar -h <path-to-hgr> -k <# blocks> -e <imbalance (e.g. 0.03)> -o km1 -m direct -p ../../../config/km1_direct_kway_sea17.ini

To start KaHyPar in direct k-way mode (KaHyPar-K) optimizing the (connectivity - 1) objective run:

./KaHyPar -h <path-to-hgr> -k <# blocks> -e <imbalance (e.g. 0.03)> -o km1 -m direct -p ../../../config/km1_direct_kway_alenex17.ini

To start KaHyPar in recursive bisection mode (KaHyPar-R) optimizing the cut-net objective run:

./KaHyPar -h <path-to-hgr> -k <# blocks> -e <imbalance (e.g. 0.03)> -o cut -m recursive -p ../../../config/cut_rb_alenex16.ini

All preset parameters can be overwritten by using the corresponding command line options.

Using the Library Interfaces

The C-Style Interface

We provide a simple C-style interface to use KaHyPar as a library. The library can be built and installed via

make install.library

and can be used like this:

#include <memory>
#include <vector>
#include <iostream>

#include <libkahypar.h>

int main(int argc, char* argv[]) {

kahypar_context_t* context = kahypar_context_new();
kahypar_configure_context_from_file(context, "/path/to/config.ini");

const kahypar_hypernode_id_t num_vertices = 7;
const kahypar_hyperedge_id_t num_hyperedges = 4;

std::unique_ptr<kahypar_hyperedge_weight_t[]> hyperedge_weights = std::make_unique<kahypar_hyperedge_weight_t[]>(4);

// force the cut to contain hyperedge 0 and 2
hyperedge_weights = 1;  hyperedge_weights = 1000;
hyperedge_weights = 1;  hyperedge_weights = 1000;

std::unique_ptr<size_t[]> hyperedge_indices = std::make_unique<size_t[]>(5);

hyperedge_indices = 0; hyperedge_indices = 2;
hyperedge_indices = 6; hyperedge_indices = 9;
hyperedge_indices = 12;

std::unique_ptr<kahypar_hyperedge_id_t[]> hyperedges = std::make_unique<kahypar_hyperedge_id_t[]>(12);

// hypergraph from hMetis manual page 14
hyperedges = 0;  hyperedges = 2;
hyperedges = 0;  hyperedges = 1;
hyperedges = 3;  hyperedges = 4;
hyperedges = 3;  hyperedges = 4;
hyperedges = 6;  hyperedges = 2;
hyperedges = 5; hyperedges = 6;

const double imbalance = 0.03;
const kahypar_partition_id_t k = 2;

kahypar_hyperedge_weight_t objective = 0;

std::vector<kahypar_partition_id_t> partition(num_vertices, -1);

kahypar_partition(num_vertices, num_hyperedges,
imbalance, k,
/*vertex_weights */ nullptr, hyperedge_weights.get(),
hyperedge_indices.get(), hyperedges.get(),
&objective, context, partition.data());

for(int i = 0; i != num_vertices; ++i) {
std::cout << i << ":" << partition[i] << std::endl;
}

kahypar_context_free(context);
}

To compile the program using g++ run:

g++ -std=c++14 -DNDEBUG -O3 -I/usr/local/include -L/usr/local/lib -lkahypar -L/path/to/boost/lib -I/path/to/boost/include -lboost_program_options program.cc -o program

To remove the library from your system use the provided uninstall target:

make uninstall-kahypar

The Julia Interface

Thanks to Jordan Jalving (@jalving) KaHyPar now also offers a Julia interface, which can currently be found here: jalving/KaHyPar.jl.

The corresponding dependency can be installed via:

using Pkg
Pkg.test("KaHyPar")

After that, you can use KaHyPar to partition your hypergraphs like this:

using KaHyPar
using SparseArrays

I = [1,3,1,2,4,5,4,5,7,3,6,7]
J = [1,1,2,2,2,2,3,3,3,4,4,4]
V = Int.(ones(length(I)))

A = sparse(I,J,V)

h = KaHyPar.hypergraph(A)

KaHyPar.partition(h,2,configuration = :edge_cut)

KaHyPar.partition(h,2,configuration = :connectivity)

KaHyPar.partition(h,2,configuration = joinpath(@__DIR__,"../src/config/km1_direct_kway_sea18.ini"))

Bug Reports

We encourage you to report any problems with KaHyPar via the github issue tracking system of the project.

Licensing

KaHyPar is free software provided under the GNU General Public License (GPLv3). For more information see the COPYING file. We distribute this framework freely to foster the use and development of hypergraph partitioning tools. If you use KaHyPar in an academic setting please cite the appropriate paper. If you are interested in a commercial license, please contact me.

// KaHyPar-R
@inproceedings{shhmss2016alenex,
author    = {Sebastian Schlag and
Vitali Henne and
Tobias Heuer and
Henning Meyerhenke and
Peter Sanders and
Christian Schulz},
title     = {k-way Hypergraph Partitioning via \emph{n}-Level Recursive
Bisection},
booktitle = {18th Workshop on Algorithm Engineering and Experiments, (ALENEX 2016)},
pages     = {53--67},
year      = {2016},
}

// KaHyPar-K
@inproceedings{ahss2017alenex,
author    = {Yaroslav Akhremtsev and
Tobias Heuer and
Peter Sanders and
Sebastian Schlag},
title     = {Engineering a direct \emph{k}-way Hypergraph Partitioning Algorithm},
booktitle = {19th Workshop on Algorithm Engineering and Experiments, (ALENEX 2017)},
pages     = {28--42},
year      = {2017},
}

// KaHyPar-CA
@inproceedings{hs2017sea,
author    = {Tobias Heuer and
Sebastian Schlag},
title     = {Improving Coarsening Schemes for Hypergraph Partitioning by Exploiting Community Structure},
booktitle = {16th International Symposium on Experimental Algorithms, (SEA 2017)},
pages     = {21:1--21:19},
year      = {2017},
}

// KaHyPar-MF
@inproceedings{heuer_et_al:LIPIcs:2018:8936,
author ={Tobias Heuer and Peter Sanders and Sebastian Schlag},
title =,
booktitle ={17th International Symposium on Experimental Algorithms  (SEA 2018)},
pages ={1:1--1:19},
year ={2018}
}

// KaHyPar-E (EvoHGP)
@inproceedings{Andre:2018:MMH:3205455.3205475,
author = {Robin Andre and Sebastian Schlag and Christian Schulz},
title = {Memetic Multilevel Hypergraph Partitioning},
booktitle = {Proceedings of the Genetic and Evolutionary Computation Conference},
series = {GECCO '18},
year = {2018},
pages = {347--354},
numpages = {8}
}

KaHyPar-MF integrates implementations of the BK and incremental breadth first search (IBFS) maximum flow algorithm into the framework (see /external_tools/maximum_flow/). The BK algorithm has been described in

“An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision.”
Yuri Boykov and Vladimir Kolmogorov.
In IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), September 2004.

The IBFS algorithm can be used for research purposes only and is described in

“Faster and More Dynamic Maximum Flow by Incremental Breadth-First Search”
Andrew V. Goldberg, Sagi Hed, Haim Kaplan, Pushmeet Kohli, Robert E. Tarjan, and Renato F. Werneck
In Proceedings of the 23rd European conference on Algorithms, ESA'15, 2015

"Maximum flows by incremental breadth-first search"
Andrew V. Goldberg, Sagi Hed, Haim Kaplan, Robert E. Tarjan, and Renato F. Werneck.
In Proceedings of the 19th European conference on Algorithms, ESA'11, 2011.

Contributing

If you are interested in contributing to the KaHyPar framework feel free to contact me or create an issue on the issue tracking system.