148 lines
6.6 KiB
Markdown
148 lines
6.6 KiB
Markdown
# Maximum Weighted Matching
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This repository contains implementations of maximum weighted matching for general graphs.
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In graph theory, a _matching_ is a subset of edges that does not use any vertex more than once.
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For an edge-weighted graph, a _maximum weight matching_ is a matching that achieves
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the largest possible sum of weights of matched edges.
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The code in this repository is based on a variant of the blossom algorithm that runs in
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_O(n*m*log(n))_ steps.
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See the file [Algorithm.md](doc/Algorithm.md) for a detailed description.
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## Python
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The folder [python/](python/) contains a Python 3 package `mwmatching` that implements
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maximum weighted matching.
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The Python code is self-contained -- it has no dependencies outside the standard library.
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To use the package, set your `PYTHONPATH` to the location of the package `mwmatching`.
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Alternatively, you can install the package into your Python environment by running
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```bash
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cd python
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pip install .
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```
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Using the algorithm is easy.
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You describe the input graph by listing its edges.
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Each edge is represented as a pair of vertex indices and the weight of the edge.
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The example below finds a matching in a graph with 5 vertices and 5 edges.
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The maximum weight matching contains two edges and has total weight 11.
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```python
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from mwmatching import maximum_weight_matching
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edges = [(0, 1, 3), (1, 2, 8), (1, 4, 6), (2, 3, 5), (2, 4, 7)]
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matching = maximum_weight_matching(edges)
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print(matching) # prints [(1, 4), (2, 3)]
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```
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## C++
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The folder [cpp/](cpp/) contains a header-only C++ implementation of maximum weighted matching.
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**NOTE:**
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The C++ code currently implements a slower algorithm that runs in _O(n<sup>3</sup>)_ steps.
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I plan to eventually update the C++ code to implement the faster _O(n*m*log(n))_ algorithm.
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The C++ code is self-contained and can easily be linked into an application.
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It is also reasonably efficient.
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For serious use cases, [LEMON](http://lemon.cs.elte.hu/trac/lemon) may be a better choice.
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LEMON is a C++ library that provides a very fast and robust implementation of
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maximum weighted matching and many other graph algorithms.
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To my knowledge, it is the only free software library that provides a high-quality
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matching algorithm.
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## Repository structure
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```
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python/
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mwmatching/ : Python package for maximum weight matching
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__init__.py
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algorithm.py : Algorithm implementation
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datastruct.py : Internal data structures
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tests/
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test_algorithm : Unit tests for the algorithm
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test_datastruct.py : Unit tests for data structures
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run_matching.py : Command-line program to run the matching algorithm
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cpp/
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mwmatching.h : C++ implementation of maximum weight matching
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test_mwmatching.cpp : Unit tests
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run_matching.cpp : Command-line program to run the matching algorithm
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tests/
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generate/ : Python programs to generate graphs
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graphs/ : Collection of graphs and matching outputs
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lemon/ : C++ program to run LEMON's matching algorithm
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run_test.py : Command-line program to test/benchmark matching solvers
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doc/
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Algorithm.md : Description of the algorithm
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```
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## Status of this software
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The code is finished and tested and it works.
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I ran extensive tests on random graphs and compared the results against another solver.
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It seems to me that most bugs would have been found through these tests.
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But it is definitely still possible that I overlooked something.
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This repository is just a side-project for me.
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I don't have any plans to develop this into a polished, full-featured, supported software package.
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## Credits
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The code in this repository was written by me, but it implements an algorithm that has been known
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for decades.
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This algorithm is entirely the work of other people.
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I did not invent any part of it, and deserve no credit for it.
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My implementation follows the description of the algorithm as it appears in the paper
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"Efficient Algorithms for Finding Maximum Matching in Graphs" by Zvi Galil (1986).
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However, earlier versions of the algorithm were invented and improved by several other scientists.
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See the file [Algorithm.md](doc/Algorithm.md) for links to the most important papers.
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I used some ideas from the source code of the `MaxWeightedMatching` class in
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[LEMON](http://lemon.cs.elto.hu/trac/lemon):
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the technique to implement lazy updates of vertex dual variables,
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and the approach to re-use alternating trees after augmenting the matching.
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I used Fortran programs `hardcard.f`, `t.f` and `tt.f` by R. B. Mattingly and N. Ritchey
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to generate test graphs.
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These programs are part of the DIMACS Network Flows and Matching implementation challenge.
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They can be found in the
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[DIMACS Netflow archive](http://archive.dimacs.rutgers.edu/pub/netflow/).
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To check the correctness of my results, I used other maximum weight matching solvers:
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the `MaxWeightedMatching` module in LEMON,
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and the program
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[`wmatch`](http://archive.dimacs.rutgers.edu/pub/netflow/matching/weighted/solver-1/)
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by Edward Rothberg.
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## License
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The following license applies to the software in this repository, excluding the folder `doc`.
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This license is sometimes called the MIT License or the Expat License:
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> Copyright (c) 2023-2024 Joris van Rantwijk
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>
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> Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
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>
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> The above copyright notice and this permission notice (including the next paragraph) shall be included in all copies or substantial portions of the Software.
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>
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>THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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The following applies to the documentation in the folder [doc](./doc/):
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> Written in 2023 by Joris van Rantwijk. <br>
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> This work is licensed under a [Creative Commons Attribution-NoDerivatives 4.0 International License](https://creativecommons.org/licenses/by-nd/4.0/).
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