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Solutions to Postman graph optimization problems: Chinese and Rural Postman problems

Project description

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Note to those reading this on PyPI: For a better reading experience, checkout the README on GitHub here. GitHub and PyPI are not cooperating on rendering SVGs.

1   Contents

This package contains implementations to solve the suite of Postman Problems from graph theory.

Algorithms currently implemented:

  • Chinese Postman
  • Rural Postman (partial)

The Rural Postman implementation is currently restricted to a subset of problems. Specifically to graphs where the required edges form a single connected component when the optional edges are removed.

2   Install

2.1   Basic

Option 1. Install from PyPI: Stable release.

pip install postman_problems

Option 2. Install from GitHub: As this project develops, GitHub will have the most recent features, but no guarantees they’ll be stable.

  1. Clone the repo.

    git clone https://github.com/brooksandrew/postman_problems.git
    cd postman_problems
    
  2. Install with pip. Builds are tested on Python 2.7, 3.3, 3.4, 3.5, 3.6.

    pip install .
    

2.2   Viz

postman_problems leverages Graphviz for visualization which unlocks more robust visualizations than just NetworkX and matplotlib. However, this also comes with several dependencies. These are managed separately from the base package, so users can optimize graphs to their heart’s content unencumbered from the weight and hassle of installing viz dependencies, if they so choose.

  1. Install optional Python visualization libraries.

    pip install postman_problems[viz]
    
  2. Install Graphviz. You need this underlying software application in addition to the graphviz python package which wraps around it. Checkout the Graphviz Download page for the best way to install on your OS.

    For Mac, this should be as easy as:

    brew install graphviz
    

    For Linux,

    sudo apt-get install graphviz
    

    These are the installs I’m currently using on my builds for the tests on TravisCI. YMMV. For Windows users and for those where these methods fail, I defer to the Graphviz download docs.

3   Usage

3.1   CLI

The easiest way to start is with the command line using the entry points installed with this package: chinese_postman and rural_postman.

3.1.1   Arguments: edgelist

There are several optional command line arguments, but the only one required is --edgelist:

Description: Filename of edgelist. Expected to be comma delimited text file readable with pandas.read_csv. The first two columns should be the “from” and “to” node names. Additional columns can be provided for edge attributes. The first row should be the edge attribute names.

A note on some edge attributes:

  • required: must be provided for the RPP. 0 is used for optional edges, 1 for required.
  • distance: default edge attribute used for shortest path computations. Can be overridden with edge_weight.
  • id: recommended to not include, but will be used if provided. This will be generated automatically to assist with computation of parallel edges. If provided, it should be unique to ensure stable computation.

3.1.2   Arguments: others

For the complete list of optional arguments run one of the following:

chinese_postman --help
rural_postman --help

The big ones are --viz and --animation, which when present will create the static (single visualization) and animation of the postman problem solution. Most of the other arguments modify the default values used for the visualizations.

3.1.3   Simple example

Below we solve the CPP on the Seven Bridges of Konigsberg network. The edgelist is provided in this repo, but you can swap this out for any comma delimited text file where the first two columns represent the node pairs in your network.

chinese_postman --edgelist postman_problems/examples/seven_bridges/edgelist_seven_bridges.csv

If the chinese_postman entry point is not working for whatever reason, you can run the script directly with:

python postman_problems/chinese_postman.py --edgelist postman_problems/examples/seven_bridges/edgelist_seven_bridges.csv

You should see output that describes the CPP route solution (Eulerian circuit through each edge). Something like this:

('A', 'C', 1, {'trail': 'd', 'distance': 10, 'id': 3})
('C', 'D', 0, {'trail': 'g', 'distance': 3, 'id': 6, 'augmented': True})
('D', 'C', 0, {'trail': 'g', 'distance': 3, 'id': 6})
('C', 'A', 0, {'trail': 'c', 'distance': 2, 'id': 2})
('A', 'D', 0, {'trail': 'e', 'distance': 1, 'id': 4})
('D', 'B', 0, {'trail': 'f', 'distance': 9, 'id': 5})
('B', 'A', 0, {'trail': 'a', 'distance': 3, 'id': 0, 'augmented': True})
('A', 'B', 1, {'trail': 'b', 'distance': 5, 'id': 1})
('B', 'A', 0, {'trail': 'a', 'distance': 3, 'id': 0})

The first two values of each tuple are the “from” and the “to” node respectively for each edge in the circuit.

The third value indicates the key of the edge on the MultiGraph. These will be 0 unless there are parallel edges.

The fourth value contains the edge attributes for each edge walked. These are mostly grabbed from the starting graph, with two exceptions:

  • augmented is added to edges after their first walk (double backing… the thing we want to minimize)
  • id is generated to aid computation in the case of parallel edges. This can generally be ignored.

A summary report of the solution should be printed. Something like this:

Solution summary stats:
distance_walked : 39
distance_doublebacked : 6
distance_walked_once : 33
distance_walked_optional : 0
distance_walked_required : 39
edges_walked : 9
edges_doublebacked : 2
edges_walked_once : 7
edges_walked_optional : 0
edges_walked_required : 9

3.2   Python

The postman solvers are modules that can also be imported and run within a Python environment. This might interest you if solving the CPP/RPP is just one step in your problem, you’d like to poke and prod at the output, or you’d like to tweak the visualization or optimization parameters beyond what’s provided from the CLI.

The snippet below should produce exactly the same output as printed above in CLI.

from postman_problems.solver import cpp
from postman_problems.stats import calculate_postman_solution_stats

# find CPP solution
circuit, graph = cpp(edgelist_filename='postman_problems/examples/seven_bridges/edgelist_seven_bridges.csv', start_node='D')

# print solution route
for e in circuit:
    print(e)

# print solution summary stats
for k, v in calculate_postman_solution_stats(circuit).items():
    print(k, v)

4   Examples

Three examples are included in postman_problems which demonstrate end-to-end usage: raw edgelist & nodelist => optimization and visualization.

Examples are added as entry points and pre-configured arguments, so they can be executed with the single commands below.

Note, the visualization step will write a GIF and a series of PNGs to your filesystem. The paths are locked into postman_problems/examples/[example_name]/output/.

An expected exception will be thrown if you don’t have the visualization dependencies. But no worries, the output is prepackaged into /examples and images are embedded below.

Each example will produce the following file types:

  • cpp_graph: representation of cpp_graph.svg in the DOT graph description language. This is provided mostly for reference, or for tweaking.
  • cpp_graph.svg: static image with edge attributes annotating the walk sequence.
  • cpp_graph.gif: animation highlighting each edge in the Euler circuit (solution route) as it’s walked.
  • png/img*.png: PNGs generated for each frame of the GIF (omitted from package, but will hit your filesystem when you run the examples).

4.1   CPP: Seven Bridges of Konigsberg

The Seven Bridges of Konigsberg is rather simple network with just 4 nodes and 7 edges. Although small, it does contain 2 parallel edges which introduce some complexity to the CPP computation.

This was the graph with which Leonhard Euler famously postulated in 1736 that there exists a path which visits each edge exactly once if all nodes have even degree. Although this wasn’t proven until the 1870s by Carl Hierholzer, Euler was right and this property is a key part of solving the Postman Problems.

This contrived example has been bundled and parameterized into a script that can be run with:

chinese_postman_seven_bridges

The example can also be run using the verbose method below which allows you to parameterize more pieces. Many of the options provided below are defaults and can be excluded in practice. They are included here simply to convey what the possibilities are:

chinese_postman --edgelist postman_problems/examples/seven_bridges/edgelist_seven_bridges.csv \
--viz \
--viz_filename 'postman_problems/examples/seven_bridges/output/cpp_graph.svg' \
--viz_engine 'dot' \
--animation \
--animation_filename 'postman_problems/examples/seven_bridges/output/cpp_graph.gif' \
--animation_images_dir 'postman_problems/examples/seven_bridges/output/img' \
--animation_engine 'dot' \
--animation_format 'png' \
--fps 2

base_cpp_graph.svg: This is the starting graph. Edges are annotated by distance.

./postman_problems/examples/seven_bridges/output/base_cpp_graph.svg

cpp_graph.svg: Edges are annotated with the order in which they are walked, starting at 0. Edges walked more than once are annotated by a sequence of numbers (walk order) and bolded.

./postman_problems/examples/seven_bridges/output/cpp_graph.svg

cpp_graph.gif: The nodes and edges in red indicate the current direction.

./postman_problems/examples/seven_bridges/output/cpp_graph.gif

cpp_graph: dot representation of the graph is also provided. This is mostly for reference, but in rare cases you may want to tweak graphviz parameters directly here.

graph {
    graph [forcelabels=true "strict"=false]
    C [label=C]
    D [label=D]
    A [label=A]
    B [label=B]
            C -- D [label=9 decorate=true distance=3 id=6 penwidth=1 trail=g]
            C -- A [label="6, 8" augmented=True decorate=true distance=2 id=2 penwidth=4 trail=c]
            C -- A [label=7 decorate=true distance=10 id=3 penwidth=1 trail=d]
            D -- A [label="0, 3" augmented=True decorate=true distance=1 id=4 penwidth=4 trail=e]
            D -- B [label=4 decorate=true distance=9 id=5 penwidth=1 trail=f]
            A -- B [label="2, 5" augmented=True decorate=true distance=3 id=0 penwidth=4 trail=a]
            A -- B [label=1 decorate=true distance=5 id=1 penwidth=1 trail=b]
}

4.2   RPP: Star

This is a simple example that demonstrates the power of the RPP vs CPP.

Run with:

rural_postman_star

base_rpp_graph.svg: Required edges are solid. Optional edges are dotted. Simply showing the edge distances here.

./postman_problems/examples/star/output/base_rpp_graph.svg

cpp_graph_req.svg: If we solve this with the CPP and only only consider the required edges, we get the pretty inefficient solution below doubling back on every single edge.

./postman_problems/examples/star/output/cpp_graph_req.svg

cpp_graph_opt.svg: If we recognize the optional edges, and apply the the CPP again (where the optional edges are treated as required edges), we get a slightly better solution:

./postman_problems/examples/star/output/cpp_graph_opt.svg

rpp_graph.svg: When we recognize the optional edges as truly optional and employ the RPP, we get the optimal solution where we walk all required edges exactly once and only use a subset of optional edges:

./postman_problems/examples/star/output/rpp_graph.svg

rpp_graph.gif: Same information as above, but in an animation… because flashy moving pictures are fun.

https://gist.githubusercontent.com/brooksandrew/d24560e674fccd1ab78f9d2588769e86/raw/4e477121ea698431ce294c6e1b17ad7e415eb396/rpp_star_example_for_postman_problems.gif

4.3   RPP: Sleeping Giant

This example is near and dear to my heart and the motivation for this project in the first place.

Sleeping Giant is a state park near my hometown in Hamden CT with a little challenge called the Giant Master Program. Those who hike every trail (see trail map) are awarded the honor of Giantmaster Marathoner and earn themselves a spot on the Giantmaster roster and the glory of a red highlight on their name.

That’s all I’ll say here. I wrote more about the personal motivation and Sleeping Giant specific data/problem in a DataCamp tutorial which also helped motivate this project.

Run this example with:

rural_postman_sleeping_giant

postman_problems/examples/sleeping_giant/rpp_graph.svg:

The optional edges are marked with a dotted line. You’ll notice that not all of them are utilized (no edge label annotating their order in the route).

./postman_problems/examples/sleeping_giant/output/rpp_graph.svg

postman_problems/examples/sleeping_giant/rpp_graph.gif (omitted from package due to size): Can be viewed here.

Here are the solution summary stats.

RPP Solution summary stats:

Solution summary stats:
distance_walked : 32.119999999999976
distance_doublebacked : 6.11
distance_walked_once : 26.009999999999977
distance_walked_optional : 0.68
distance_walked_required : 31.439999999999976
edges_walked : 151
edges_doublebacked : 30
edges_walked_once : 121
edges_walked_optional : 2
edges_walked_required : 149

A CPP example is also provided with entry point chinese_postman_sleeping_giant. The solution is very similar, so it is omitted here.

For a base of comparison on RPP vs CPP, selected stats are printed below for the CPP. the RPP shortens the CPP solution route by about 1 mile.

CPP Solution summary stats:

distance_walked : 33.24999999999998
distance_doublebacked : 7.240000000000001
distance_walked_once : 26.009999999999977
edges_walked : 155
edges_doublebacked : 34
edges_walked_once : 121

5   Developers

If you’d like to fork or contribute directly to this project (PRs welcome), or simply want run the tests, here’s how:

  1. Clone/Fork repo

  2. Full install with test and viz dependencies.

    pip install .[test,viz]
    

    Or do an editable install from the beginning. This is my typical approach when developing.

    pip install -e .[test,viz]
    
  3. Run tests

    python -m pytest
    pytest --cov
    

    Some tests take quite a while to run. Namely the examples that write visualizations to the filesystem for large networks.

    As I have limited patience while developing, but am too cautious to drop them completely, I’ve kept and marked them with the @slow and @long decorators. conftest.py is configured to exclude them by default with a simple run of pytest or python -m pytest, but the full test suite can be run by:

    python -m pytest --runslow
    pytest --cov --runslow
    

6   Release Notes

Checkout the release notes in Gitub here.

If I’m doing a good job of keeping PyPI updated, each release should also be available here.

7   License

Released under the MIT License (see LICENSE.txt).

Copyright (C) 2017 Andrew Brooks.

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