https://github.com/filipporanza/graph-coloring
A Branch-and-Price implementation of the graph coloring problem using SCIP
https://github.com/filipporanza/graph-coloring
grahp graph-coloring mathematical-programming milp scip
Last synced: 6 months ago
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A Branch-and-Price implementation of the graph coloring problem using SCIP
- Host: GitHub
- URL: https://github.com/filipporanza/graph-coloring
- Owner: FilippoRanza
- Created: 2022-05-13T14:32:38.000Z (over 3 years ago)
- Default Branch: main
- Last Pushed: 2022-06-06T13:08:56.000Z (over 3 years ago)
- Last Synced: 2025-04-14T22:56:32.018Z (6 months ago)
- Topics: grahp, graph-coloring, mathematical-programming, milp, scip
- Language: C
- Homepage:
- Size: 45.9 KB
- Stars: 0
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# graph-coloring
## MILP Model formulation
The Graph Coloring problem, also known as Vertex Coloring Problem, aims
to color each vertex of a given graph with the minimal number of different colors,
given that two adjacent vertex cannot have the same color. See [the wikipedia article](https://en.wikipedia.org/wiki/Graph_coloring) for
more information.
### Base Model
$$
\begin{align*}
\min& \sum_{c \in C} y_c\\
\text{subject to}&\\
&\sum_{c \in C} x_{ic} = 1\ \ \ \ \forall i \in V\\
&x_{ic} + x_{jc} \le 1\ \ \ \ \forall(i, j) \in E, \forall c \in C\\
&x_{ic} \le y_{ic} \ \ \ \ \forall i \in V, \forall c \in C\\
&x_{ic} \in \lbrace0, 1\rbrace \ \ \ \ \forall i \in V, \forall c \in C\\
&y_{c} \in \lbrace0, 1\rbrace\ \ \ \ \forall c \in C\\
\end{align*}
$$
### Branch and Price Model
#### Master Problem
$$
\begin{align*}
\min& \sum_{p \in P} \lambda_p\\
\text{subject to}&\\
&\sum_{p \in P} a_{ip}\lambda_p = 1\ \ \ \ \forall i \in V\\
&\lambda_p \in \lbrace0, 1\rbrace\ \ \ \ \forall p \in P\\
\end{align*}
$$
#### Pricing Problem
$$
\begin{align*}
\min&\ \ 1 - \sum_{i \in V} \pi_i x_i\\
\text{subject to}&\\
&x_{i} + x_{j} \le 1\ \ \ \ \forall(i, j) \in E\\
&x_{i} \in \lbrace0, 1\rbrace \ \ \ \ \forall i \in V
\end{align*}
$$
## Compile
In order to compile this tool one needs cmake and SCIP.
This tools has been developed and tested using SCIP 8: it may work
with previous versions but I cannot be sure. cmake scripts are
configured to work with a system installation of SCIP: you may need to
modify the configuration if you want or need to use a static linked SCIP library.
To compile *graph-coloring*
```bash
# Configure build environment
cmake -S . -B build/
# Compile source code
cmake --build build/
# Run unit tests
ctest --test-dir build/
```
Once the compiling and testing procedures are done you will find inside
the **build/** directory two executables:
+ *bnp-graph-coloring* : branch and price implementation of the graph coloring problem;
+ *graph-coloring* : arc based implementation of the graph coloring problem.
Note: on your platform you may find similar names but with extensions like *.exe*.
### Usage
The two generated executables takes as input, from the command line, a single
ASCII DIMACS file. The parser is trivial and assumes that the input file is
correctly formatted. The parser does not support binary files.
The two generated executables works in the same way so, for brevity, I will show
only the usage of *bnp-graph-coloring*.
```bash
./bnp-graph-coloring your-graph.col
```
If you do not provide a graph file as argument the tool will get the graph
specification from standard input.