baldor 0.0.3

Estimating the Minimum Vertex Cover for an undirected graph encoded in DIMACS format and stored in a file.

Baldor: Minimum Vertex Cover Solver

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The Minimum Vertex Cover Problem

The Minimum Vertex Cover (MVC) problem is a classic optimization problem in computer science and graph theory. It involves finding the smallest set of vertices in a graph that covers all edges, meaning at least one endpoint of every edge is included in the set.

Formal Definition

Given an undirected graph $G = (V, E)$, a vertex cover is a subset $V' \subseteq V$ such that for every edge $(u, v) \in E$, at least one of $u$ or $v$ belongs to $V'$. The MVC problem seeks the vertex cover with the smallest cardinality.

Importance and Applications

• Theoretical Significance: MVC is a well-known NP-hard problem, central to complexity theory.
• Practical Applications:
  • Network Security: Identifying critical nodes to disrupt connections.
  • Bioinformatics: Analyzing gene regulatory networks.
  • Wireless Sensor Networks: Optimizing sensor coverage.

Related Problems

• Maximum Independent Set: The complement of a vertex cover.
• Set Cover Problem: A generalization of MVC.

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Problem Statement

Input: A Boolean Adjacency Matrix $M$.

Answer: Find a Minimum Vertex Cover.

Example Instance: 5 x 5 matrix

	c1	c2	c3	c4	c5
r1	0	0	1	0	1
r2	0	0	0	1	0
r3	1	0	0	0	1
r4	0	1	0	0	0
r5	1	0	1	0	0

The input for undirected graph is typically provided in DIMACS format. In this way, the previous adjacency matrix is represented in a text file using the following string representation:

p edge 5 4
e 1 3
e 1 5
e 2 4
e 3 5

This represents a 5x5 matrix in DIMACS format such that each edge $(v,w)$ appears exactly once in the input file and is not repeated as $(w,v)$. In this format, every edge appears in the form of

e W V

where the fields W and V specify the endpoints of the edge while the lower-case character e signifies that this is an edge descriptor line.

Example Solution:

Vertex Cover Found 1, 2, 3: Nodes 1, 2, and 3 constitute an optimal solution.

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Compile and Environment

Prerequisites

• Python ≥ 3.10

Installation

pip install baldor

Execution

1. Clone the repository:

   git clone https://github.com/frankvegadelgado/baldor.git
   cd baldor
   

2. Run the script:

   solve -i ./benchmarks/testMatrix1
   

   utilizing the solve command provided by Baldor's Library to execute the Boolean adjacency matrix baldor\benchmarks\testMatrix1. The file testMatrix1 represents the example described herein. We also support .xz, .lzma, .bz2, and .bzip2 for compressed text files.

   Example Output:

   testMatrix1: Vertex Cover Found 1, 2, 3
   

   This indicates nodes 1, 2, 3 form a vertex cover.

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Vertex Cover Size

Use the -c flag to count the nodes in the vertex cover:

solve -i ./benchmarks/testMatrix2 -c

Output:

testMatrix2: Vertex Cover Size 5

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Command Options

Display help and options:

solve -h

Output:

usage: solve [-h] -i INPUTFILE [-a] [-b] [-c] [-v] [-l] [--version]

Estimating the Minimum Vertex Cover for an undirected graph encoded in DIMACS format and stored in a file.

options:
  -h, --help            show this help message and exit
  -i INPUTFILE, --inputFile INPUTFILE
                        input file path
  -a, --approximation   enable comparison with a polynomial-time approximation approach within a factor of at most 2
  -b, --bruteForce      enable comparison with the exponential-time brute-force approach
  -c, --count           calculate the size of the vertex cover
  -v, --verbose         anable verbose output
  -l, --log             enable file logging
  --version             show program's version number and exit

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Batch Execution

Batch execution allows you to solve multiple graphs within a directory simultaneously.

To view available command-line options for the batch_solve command, use the following in your terminal or command prompt:

batch_solve -h

This will display the following help information:

usage: batch_solve [-h] -i INPUTDIRECTORY [-a] [-b] [-c] [-v] [-l] [--version]

Estimating the Minimum Vertex Cover for all undirected graphs encoded in DIMACS format and stored in a directory.

options:
  -h, --help            show this help message and exit
  -i INPUTDIRECTORY, --inputDirectory INPUTDIRECTORY
                        Input directory path
  -a, --approximation   enable comparison with a polynomial-time approximation approach within a factor of at most 2
  -b, --bruteForce      enable comparison with the exponential-time brute-force approach
  -c, --count           calculate the size of the vertex cover
  -v, --verbose         anable verbose output
  -l, --log             enable file logging
  --version             show program's version number and exit

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Code

• Python implementation by Frank Vega.

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License

• MIT License.