Alonso: Approximate Vertex Cover Solver

This work builds upon Challenging the Unique Games Conjecture.

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.

Overview of the Algorithm and Its Running Time

The find_vertex_cover algorithm approximates a minimum vertex cover for an undirected graph $G = (V, E)$ by partitioning its edges into two claw-free subgraphs using the Burr-Erdős-Lovász (1976) method, computing exact vertex covers for these subgraphs with the Faenza, Oriolo, and Stauffer (2011) approach, and recursively refining the solution on residual edges. This process prevents the ratio from reaching 2, leveraging overlap between subgraphs and minimal additions in recursion. The algorithm begins by cleaning the graph (removing self-loops and isolates in $\mathcal{O}(n + m)$), checking for claw-free in $\mathcal{O}(m \cdot \Delta)$ where $\Delta$ is the maximum degree, partitions edges in $\mathcal{O}(n^3)$, computes vertex covers in $\mathcal{O}(n^3)$ per subgraph (total $\mathcal{O}(n^3)$), merges covers in $\mathcal{O}(n \cdot \log n)$, and constructs the residual graph in $\mathcal{O}(m)$. The recursion depth never exceeds a small constant, most commonly 2. This yields a total runtime of $\mathcal{O}(n^3)$, per a constant time of recursion levels.

Problem Statement

Input: A Boolean Adjacency Matrix $M$.

Answer: Find a Minimum Vertex Cover.

Example Instance: 5 x 5 matrix

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 3, 4, 5: Nodes 3, 4, and 5 constitute an optimal solution.

Compile and Environment

Prerequisites

• Python ≥ 3.12

Installation

pip install alonso

Execution

1. Clone the repository:

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

2. Run the script:

   mvc -i ./benchmarks/testMatrix1
   

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

   Example Output:

   testMatrix1: Vertex Cover Found 3, 4, 5
   

   This indicates nodes 3, 4, 5 form a vertex cover.

Vertex Cover Size

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

mvc -i ./benchmarks/testMatrix2 -c

Output:

testMatrix2: Vertex Cover Size 5

Command Options

Display help and options:

mvc -h

Output:

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

Compute an Approximate Vertex Cover for undirected graph encoded in DIMACS format.

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

Batch Execution

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

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

batch_mvc -h

This will display the following help information:

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

Compute an Approximate 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

Testing Application

A command-line utility named test_mvc is provided for evaluating the Algorithm using randomly generated, large sparse matrices. It supports the following options:

usage: test_mvc [-h] -d DIMENSION [-n NUM_TESTS] [-s SPARSITY] [-a] [-b] [-c] [-w] [-v] [-l] [--version]

The Alonso Testing Application using randomly generated, large sparse matrices.

options:
  -h, --help            show this help message and exit
  -d DIMENSION, --dimension DIMENSION
                        an integer specifying the dimensions of the square matrices
  -n NUM_TESTS, --num_tests NUM_TESTS
                        an integer specifying the number of tests to run
  -s SPARSITY, --sparsity SPARSITY
                        sparsity of the matrices (0.0 for dense, close to 1.0 for very sparse)
  -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
  -w, --write           write the generated random matrix to a file in the current directory
  -v, --verbose         anable verbose output
  -l, --log             enable file logging
  --version             show program's version number and exit

Code

• Python implementation by Frank Vega.

License

• MIT License.