hvala 0.0.1

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

Hvala: Approximate Vertex Cover Solver

This work builds upon Disproving the Unique Games Conjecture.

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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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Vertex Cover via Degree Reduction Algorithm

Algorithm Overview

The Vertex Cover via Degree Reduction Algorithm is a polynomial-time approximation algorithm that finds near-optimal vertex covers by transforming the input graph into a simpler structure where optimal solutions can be computed efficiently.

Core Approach

1. Preprocessing: Remove self-loops and isolated vertices from the input graph
2. Component Decomposition: Process each connected component independently
3. Degree Reduction: Transform each component using a novel reduction technique:
   • Replace each vertex u of degree k with k auxiliary vertices
   • Connect each auxiliary vertex to one of u's original neighbors
   • Assign weight 1/k to each auxiliary vertex
   • Resulting graph has maximum degree ≤ 1 (paths and cycles only)
4. Optimal Solving: Apply two different greedy algorithms on the reduced graph:
   • Minimum weighted dominating set algorithm
   • Minimum weighted vertex cover algorithm
5. Solution Selection: Choose the better of the two solutions
6. Extraction: Map auxiliary vertices back to original vertices

Key Innovation

The algorithm's strength lies in its dual-approach strategy: by solving both dominating set and vertex cover problems optimally on the degree-1 reduced graph and selecting the better solution, it consistently outperforms single-approach algorithms.

Performance Guarantees

Approximation Ratio

• Theoretical Bound: < 2 (strict inequality)
• Practical Performance: Often significantly better than 2, approaching optimal for many graph classes
• Comparison: Outperforms classical algorithms like the standard edge-based 2-approximation

Time Complexity

• Overall Runtime: O(|V| + |E|) - linear time
• Space Complexity: O(|V| + |E|) for storing the reduced graph

Complexity Breakdown

Phase	Time Complexity	Description
Preprocessing	`O(	V
Component Finding	`O(	V
Graph Reduction	`O(	E
Optimal Solving	`O(	V
Solution Extraction	`O(	V

Advantages

✅ Superior Approximation: Achieves approximation ratio < 2 (better than classical algorithms)

✅ Optimal Time Complexity: Linear time O(|V| + |E|) - matches the best possible for graph problems

✅ Practical Efficiency: Often produces near-optimal solutions in real-world instances

✅ Theoretical Rigor: Formal proofs guarantee correctness and performance bounds

✅ Robust Design: Handles all graph types including disconnected graphs and edge cases

Use Cases

The algorithm is particularly effective for:

• Large sparse graphs where linear time complexity is crucial
• Graphs with moderate vertex degrees where the reduction preserves structure well
• Applications requiring proven approximation guarantees with practical efficiency
• Real-time systems where predictable linear performance is essential

Implementation Notes

The algorithm requires:

• NetworkX for graph operations
• Custom greedy solvers for minimum weighted dominating set and vertex cover on degree-1 graphs
• Efficient data structures for mapping between original and auxiliary vertices

The dual-solution approach (trying both dominating set and vertex cover) is essential for achieving the < 2 approximation ratio and should not be omitted in implementations.

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Compile and Environment

Prerequisites

• Python ≥ 3.12

Installation

pip install hvala

Execution

1. Clone the repository:

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

2. Run the script:

   idemo -i ./benchmarks/testMatrix1
   

   utilizing the idemo command provided by Hvala's Library to execute the Boolean adjacency matrix hvala\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 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:

idemo -i ./benchmarks/testMatrix2 -c

Output:

testMatrix2: Vertex Cover Size 5

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Command Options

Display help and options:

idemo -h

Output:

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

Compute the 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

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Batch Execution

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

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

batch_idemo -h

This will display the following help information:

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

Compute the 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

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Testing Application

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

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

The Hvala 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

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Code

• Python implementation by Frank Vega.

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License

• MIT License.