The graph partitioning framework KaHIP -- Karlsruhe High Quality Partitioning.

The graph partitioning problem asks for a division of a graph's node set into k equally sized blocks such that the number of edges that run between the blocks is minimized. KaHIP is a family of graph partitioning programs. It includes KaFFPa (Karlsruhe Fast Flow Partitioner), which is a multilevel graph partitioning algorithm, in its variants Strong, Eco and Fast, KaFFPaE (KaFFPaEvolutionary), which is a parallel evolutionary algorithm that uses KaFFPa to provide combine and mutation operations, as well as KaBaPE which extends the evolutionary algorithm. Moreover, specialized techniques are included to partition road networks (Buffoon), to output a vertex separator from a given partition as well as techniques geared towards the efficient partitioning of social networks. Here is an overview of our framework:

Support for Python: KaHIP can now also be used in Python. See below how to do that.

Hierarchical Partitionings: KaHIP can compute hierarchial partitionings. All you have to do is specify the hierarchy and KaHIP is ready to go and does the multisection as you specified.

Node Ordering Algorithms: Many applications rely on time-intensive matrix operations, such as factorization, which can be sped up significantly for large sparse matrices by interpreting the matrix as a sparse graph and computing a node ordering that minimizes the so-called fill-in. Here, we added new algorithms to compute fill-in reduced orderings in graphs.

ILPs For Even Higher Quality: ILPs typically don't scale to large instances. We adapt them to heuristically improve a given partition. We do so by defining a much smaller model that allows us to use symmetry breaking and other techniques that make the approach scalable. We include ILPs that can be used as a post-processing step to improve high-quality partitions even further. The codes are now included in KaHIP.

TCMalloc: we added the possibility to link against TCMalloc. Depending on your system, this can yield an overall faster algorithm since it provides faster malloc operations.

Faster IO: we added an option to kaffpa (option --mmap_io) that speedsup the IO of text files significantly -- sometimes by an order of magnitude.

Added Support for Vertex and Edge Weights in ParHIP: we extended the IO functionality of ParHIP to read weighted graphs in the METIS format.

Global Multisection Mapping: we added global multisection n-to-1 process mapping algorithms. This computes better process mapping for parallel applications if information about the system hierarchy/architecture is known.

Determinism in ParHIP: we added an option to run ParHIP deterministically, i.e. two runs of ParHIP using the same seed will always return the same result.

Version Flag: we added an option to output the version that you are currently using, use the --version option of the programs.

ParHIP (Parallel High Quality Partitioning): Our distributed memory parallel partitioning techniques are designed to partition hierarchically structured networks such as web graphs or social networks.

Mapping Algorithms: Our new algorithms to map the blocks onto processors to minimize overall communication time based on hierarchical partitionings of the task graph and fast local search algorithms.

Edge Partitioning Algorithms: Our new algorithms to compute edge partitionings of graphs.

https://kahip.github.io

You can download KaHIP with the following command line:

git clone https://github.com/KaHIP/KaHIP

Before you can start, you need to install the following software packages:

• if you want to use parallel algorithms contained within the framework (e.g. ParHIP), you need OpenMPI (https://www.open-mpi.org/). If you don't want to run ParHIP, you can easily get rid of this dependency.

Once you installed the packages, just type

./compile_withcmake.sh 

In this case, all binaries, libraries and headers are in the folder ./deploy/

Note that this script detects the amount of available cores on your machine and uses all of them for the compilation process. If you don't want that, set the variable NCORES to the number of cores that you would like to use for compilation.

Alternatively use the standard cmake build process:

mkdir build
cd build 
cmake ../ -DCMAKE_BUILD_TYPE=Release     
make 
cd ..

In this case, the binaries, libraries and headers are in the folder ./build as well as ./build/parallel/parallel_src/

We also provide the option to link against TCMalloc. If you have it installed, run cmake with the additional option -DUSE_TCMALLOC=On.

By default node ordering programs are also compiled. If you have Metis installed, the build script also compiles a faster node ordering program that uses reductions before calling Metis ND. Note that Metis requires GKlib (https://github.com/KarypisLab/GKlib).

If you use the option -DUSE_ILP=On and you have Gurobi installed, the build script compiles the ILP programs to improve a given partition ilp_improve and an exact solver ilp_exact. Alternatively, you can also pass these options to ./compile_withmake.sh for example:

./compile_withcmake -DUSE_ILP=On

We also provide an option to support 64 bit edges. In order to use this, compile KaHIP with the option -D64BITMODE=On.

Lastly, we provide an option for determinism in ParHIP, e.g. two runs with the same seed will give you the same result. Note however that this option can reduce the quality of partitions, as initial partitioning algorithms do not use sophisticated memetic algorithms, but only multilevel algorithms to compute initial partitionings. ONLY use this option if you use ParHIP as a tool. Do not use this option if you want to make quality comparisons against ParHIP. To make use of this option, run

./compile_withcmake -DDETERMINISTIC_PARHIP=On

For a description of the graph format (and an extensive description of all other programs) please have a look into the manual. We give short examples here.

These programs and configurations take a graph and partition it more or less sequentially. We list here kaffpa and kaffpaE (the evolutionary framework) and their configurations. In general, the configurations are such that you can invest a lot of time into solution quality using the memetic algorithm. The memetic algorithm can also be run in parallel using MPI. In general, the more time and resources you invest, the better will be the quality of your partition. We have a lot of trade-offs, contact us if you are unsure what works best for your application. For a description of the algorithm have a look at the references that we list in the manual.

./deploy/graph_checker ./examples/rgg_n_2_15_s0.graph 

./deploy/kaffpa ./examples/rgg_n_2_15_s0.graph --k 4  --preconfiguration=strong

mpirun -n 24 ./deploy/kaffpaE ./examples/rgg_n_2_15_s0.graph --k 4  --time_limit=3600 --mh_enable_tabu_search --mh_enable_kabapE 

A large part of the project is distributed memory parallel algorithms designed for networks having a hierarchical cluster structure such as web graphs or social networks. Unfortunately, previous parallel graph partitioners originally developed for more regular mesh-like networks do not work well for complex networks. Here we address this problem by parallelizing and adapting the label propagation technique originally developed for graph clustering. By introducing size constraints, label propagation becomes applicable for both the coarsening and the refinement phase of multilevel graph partitioning. This way we exploit the hierarchical cluster structure present in many complex networks. We obtain very high quality by applying a highly parallel evolutionary algorithm to the coarsest graph. The resulting system is both more scalable and achieves higher quality than state-of-the-art systems like ParMetis or PT-Scotch.

Our distributed memory parallel algorithm can read binary files as well as standard Metis graph format files. Binary files are, in general, much more scalable than reading text files in parallel applications. The way to go here is to convert the Metis file into a binary file first (ending .bgf) and then load this one.

./deploy/graph2binary examples/rgg_n_2_15_s0.graph examples/rgg_n_2_15_s0.bgf

mpirun -n 24 ./deploy/parhip ./examples/rgg_n_2_15_s0.graph --k 4 --preconfiguration=fastmesh

mpirun -n 24 ./deploy/parhip ./examples/rgg_n_2_15_s0.bgf --k 4 --preconfiguration=fastmesh

The node separator problem asks to partition the node set of a graph into three sets A, B and S such that the removal of S disconnects A and B. We use flow-based and localized local search algorithms within a multilevel framework to compute node separators. KaHIP can also compute node separators. It can do so with a standard node separator (2-way), but it can also compute k-way node separators.

./deploy/node_separator examples/rgg_n_2_15_s0.graph

Applications such as factorization can be sped up significantly for large sparse matrices by interpreting the matrix as a sparse graph and computing a node ordering that minimizes the so-called fill-in. By applying both new and existing data reduction rules exhaustively before nested dissection, we obtain improved quality and at the same time large improvements in running time on a variety of instances. If METIS is installed, the build script also compiles the fast_node_ordering program, which runs reductions before running Metis to compute an ordering. The programs are also available through the library.

./deploy/node_ordering examples/rgg_n_2_15_s0.graph

./deploy/fast_node_ordering examples/rgg_n_2_15_s0.graph

Edge-centric distributed computations have appeared as a recent technique to improve the shortcomings of think- like-a-vertex algorithms on large scale-free networks. In order to increase parallelism on this model, edge partitioning -- partitioning edges into roughly equally sized blocks -- has emerged as an alternative to traditional (node-based) graph partitioning. We include a fast parallel and sequential split-and-connect graph construction algorithm that yields high-quality edge partitions in a scalable way. Our technique scales to networks with billions of edges, and runs efficiently on thousands of PEs.

./deploy/edge_partitioning ./examples/rgg_n_2_15_s0.graph --k 4 --preconfiguration=fast

mpirun -n 4 ./deploy/distributed_edge_partitioning ./examples/rgg_n_2_15_s0.bgf --k 4 --preconfiguration=fastsocial 

Communication and topology aware process mapping is a powerful approach to reduce communication time in parallel applications with known communication patterns on large, distributed memory systems. We address the problem as a quadratic assignment problem (QAP), and include algorithms to construct initial mappings of processes to processors as well as fast local search algorithms to further improve the mappings. By exploiting assumptions that typically hold for applications and modern supercomputer systems such as sparse communication patterns and hierarchically organized communication systems, we arrive at significantly more powerful algorithms for these special QAPs. Our multilevel construction algorithms employ perfectly balanced graph partitioning techniques and excessively exploit the given communication system hierarchy. Since v3.0 we included global multisection algorithms that directly partition the input network along the specified hierarchy to obtain an n-to-1 mapping and afterwards call 1-to-1 mapping algorithms to even further improve the mapping.

./deploy/kaffpa examples/rgg_n_2_15_s0.graph --k 256 --preconfiguration=eco --enable_mapping --hierarchy_parameter_string=4:8:8 --distance_parameter_string=1:10:100

./deploy/global_multisection examples/rgg_n_2_15_s0.graph --preconfiguration=eco  --hierarchy_parameter_string=4:3:3:3 --distance_parameter_string=1:10:100:200

We provide an ILP as well as an ILP to improve a given partition. We extend the neighborhood of the combination problem for multiple local searches by employing integer linear programming. This enables us to find even more complex combinations and hence to further improve solutions. However, out of the box those the ILPs for the problem typically do not scale to large inputs, in particular because the graph partitioning problem has a very large amount of symmetry -- given a partition of the graph, each permutation of the block IDs gives a solution having the same objective and balance. We define a much smaller graph, called model, and solve the graph partitioning problem on the model to optimality by the integer linear program. Besides other things, this model enables us to use symmetry breaking, which allows us to scale to much larger inputs. In order to compile these programs you need to run cmake in the build process above as cmake ../ -DCMAKE_BUILD_TYPE=Release -DUSE_ILP=On or run ./compile_withcmake -DUSE_ILP=On.

./deploy/ilp_improve ./examples/rgg_n_2_15_s0.graph --k 4 --input_partition=tmppartition4

./deploy/ilp_exact ./examples/example_weighted.graph  --k 3

KaHIP also offers libraries and interfaces to link the algorithms directly to your code. We explain the details of the interface in the manual. Below we list an example program that links the kahip library. This example can also be found in misc/example_library_call/.

#include <iostream>
#include <sstream>

#include "kaHIP_interface.h"


int main(int argn, char **argv) {

        std::cout <<  "partitioning graph from the manual"  << std::endl;

        int n            = 5;
        int* xadj        = new int[6];
        xadj[0] = 0; xadj[1] = 2; xadj[2] = 5; xadj[3] = 7; xadj[4] = 9; xadj[5] = 12;

        int* adjncy      = new int[12];
        adjncy[0]  = 1; adjncy[1]  = 4; adjncy[2]  = 0; adjncy[3]  = 2; adjncy[4]  = 4; adjncy[5]  = 1; 
        adjncy[6]  = 3; adjncy[7]  = 2; adjncy[8]  = 4; adjncy[9]  = 0; adjncy[10] = 1; adjncy[11] = 3; 
        
        double imbalance = 0.03;
        int* part        = new int[n];
        int edge_cut     = 0;
        int nparts       = 2;
        int* vwgt        = NULL;
        int* adjcwgt     = NULL;

        kaffpa(&n, vwgt, xadj, adjcwgt, adjncy, &nparts, &imbalance, false, 0, ECO, & edge_cut, part);

        std::cout <<  "edge cut " <<  edge_cut  << std::endl;
}

KaHIP can also be used in Python. If you want to use it in Python first run

python3 -m pip install pybind11

Then run

./compile_withcmake.sh BUILDPYTHONMODULE

to build the Python model. This will build the Python module and also put an example callkahipfrompython.py into the deploy folder. You can run this by typing the following in the deploy folder:

python3 callkahipfrompython.py 

Note that we only provide preliminary support, i.e. you may need to change some paths to Python inside the compile_withcmake file. An example can also be found below:

import kahip;

#build adjacency array representation of the graph
xadj           = [0,2,5,7,9,12];
adjncy         = [1,4,0,2,4,1,3,2,4,0,1,3];
vwgt           = [1,1,1,1,1]
adjcwgt        = [1,1,1,1,1,1,1,1,1,1,1,1]
supress_output = 0
imbalance      = 0.03
nblocks        = 2 
seed           = 0

# set mode 
#const int FAST           = 0;
#const int ECO            = 1;
#const int STRONG         = 2;
#const int FASTSOCIAL     = 3;
#const int ECOSOCIAL      = 4;
#const int STRONGSOCIAL   = 5;
mode = 0 

edgecut, blocks = kahip.kaffpa(vwgt, xadj, adjcwgt, 
                              adjncy,  nblocks, imbalance, 
                              supress_output, seed, mode)

print(edgecut)
print(blocks)

The program is licenced under MIT licence. If you publish results using our algorithms, please acknowledge our work by quoting the following paper:

@inproceedings{sandersschulz2013,
             AUTHOR = {Sanders, Peter and Schulz, Christian},
             TITLE = {{Think Locally, Act Globally: Highly Balanced Graph Partitioning}},
             BOOKTITLE = {Proceedings of the 12th International Symposium on Experimental Algorithms (SEA'13)},
             SERIES = {LNCS},
             PUBLISHER = {Springer},
             YEAR = {2013},
             VOLUME = {7933},
             PAGES = {164--175}
}

If you use our parallel partitioner ParHIP please also cite the following paper:

@inproceedings{meyerhenkesandersschulz2017,
             AUTHOR = {Meyerhenke, Henning and Sanders, Peter and Schulz, Christian},
             TITLE = {{Parallel Graph Partitioning for Complex Networks}},
             JOURNAL = {IEEE Transactions on Parallel and Distributed Systems (TPDS)},
             VOLUME = {28},
             NUMBER = {9},
             PAGES = {2625--2638},
             YEAR = {2017}
}

If you use mapping algorithm please also cite the following paper:

@inproceedings{schulztraeff2017,
             AUTHOR = {Schulz, Christian and Träff, Jesper Larsson},
             TITLE = {{Better Process Mapping and Sparse Quadratic Assignment}},
             BOOKTITLE = {Proceedings of the 16th International Symposium on Experimental Algorithms (SEA'17)},
             PUBLISHER = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik},
             VOLUME = {75},
             SERIES = {LIPIcs},
             PAGES = {4:1--4:15},
             YEAR = {2017}
}

If you use edge partitioning algorithms please also cite the following paper:

@inproceedings{edgepartitioning2019,
             AUTHOR = {Schlag, Sebastian and Schulz, Christian and Seemaier, Daniel and Strash, Darren},
             TITLE = {{Scalable Edge Partitioning}},
             BOOKTITLE = {Proceedings of the 21th Workshop on Algorithm Engineering and Experimentation (ALENEX)},
             PUBLISHER = {SIAM},
             PAGES = {211--225},
             YEAR = {2019}
}

If you use node ordering algorithms please also cite the following paper:

@article{DBLP:journals/corr/abs-2004-11315,
  author    = {Wolfgang Ost and
               Christian Schulz and
               Darren Strash},
  title     = {Engineering Data Reduction for Nested Dissection},
  journal   = {CoRR},
  volume    = {abs/2004.11315},
  year      = {2020},
  url       = {https://arxiv.org/abs/2004.11315},
  archivePrefix = {arXiv},
  eprint    = {2004.11315},
  timestamp = {Tue, 28 Apr 2020 16:10:02  0200},
  biburl    = {https://dblp.org/rec/journals/corr/abs-2004-11315.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

If you use ILP algorithms to improve a partition please also cite the following paper:

@inproceedings{DBLP:conf/wea/HenzingerN018,
  author    = {Alexandra Henzinger and
               Alexander Noe and
               Christian Schulz},
  title     = {ILP-based Local Search for Graph Partitioning},
  booktitle = {17th International Symposium on Experimental Algorithms, {SEA} 2018},
  pages     = {4:1--4:15},
  year      = {2018},
  url       = {https://doi.org/10.4230/LIPIcs.SEA.2018.4},
  doi       = {10.4230/LIPIcs.SEA.2018.4},
  series    = {LIPIcs},
  volume    = {103},
  publisher = {Schloss Dagstuhl - Leibniz-Zentrum f{\"{u}}r Informatik}
}

Yaroslav Akhremtsev

Adil Chhabra

Marcelo Fonseca Faraj

Roland Glantz

Alexandra Henzinger

Dennis Luxen

Henning Meyerhenke

Alexander Noe

Mark Olesen

Lara Ost

Ilya Safro

Peter Sanders

Hayk Sargsyan

Sebastian Schlag

Christian Schulz (maintainer)

Daniel Seemaier

Darren Strash

Jesper Larsson Träff