Capablanca: 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

	c0	c1	c2	c3	c4
r0	0	0	1	0	1
r1	0	0	0	1	0
r2	1	0	0	0	1
r3	0	1	0	0	0
r4	1	0	1	0	0

A matrix is represented in a text file using the following string representation:

00101
00010
10001
01000
10100

This represents a 5x5 matrix where each line corresponds to a row, and '1' indicates a connection or presence of an element, while '0' indicates its absence.

Example Solution:

Vertex Cover Found 0, 1, 2: Nodes 0, 1, 2 form an optimal solution.

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Our Algorithm - Polynomial Runtime

Algorithm Overview

1. Input Validation
   Ensures the input is a valid sparse adjacency matrix.

2. Graph Construction
   Converts the sparse adjacency matrix into a graph using the networkx library.

3. Component Decomposition
   Decomposes the graph into its connected components for independent processing.

4. Bipartition and Matching
   For each connected component that is a bipartite graph:

   • Find a maximum matching using an appropriate algorithm (e.g., Hopcroft-Karp).
   • Construct a vertex cover from the matching.

5. Vertex Cover Construction
   Combines the vertex covers obtained from all bipartite components.

6. Maximal Matching for Non-Bipartite Components
   For connected components that are not bipartite:

   • Find a maximal matching (not to be confused with maximum matching) using a greedy algorithm.
   • Select one endpoint for each edge in the matching, prioritizing vertices with higher degrees.

7. Iterative Processing

   • Remove the selected vertices from the graph.
   • Split the remaining graph into new connected components.
   • Repeat the process until all edges are covered.

Correctness

• Ensures all edges are covered by leveraging bipartite graph properties and maximum matchings.

Runtime Analysis

• Graph Construction: $O(|V| + |E|)$
• Maximum Matching: $O(|E| \log |V|)$ (Hopcroft-Karp algorithm)
• Maximal Matching: $O(|E|)$

Overall, the algorithm runs in polynomial time.

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Compile and Environment

Prerequisites

• Python ≥ 3.10

Installation

pip install capablanca

Execution

1. Clone the repository:

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

2. Run the script:

   cover -i ./benchmarks/testMatrix1.txt

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

   Example Output:

   testMatrix1.txt: Vertex Cover Found 0, 1, 2
   

   This indicates nodes 0, 1, 2 form a vertex cover.

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Vertex Cover Size

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

cover -i ./benchmarks/testMatrix2.txt -c

Output:

testMatrix2.txt: Vertex Cover Size 6

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Command Options

Display help and options:

cover -h

Output:

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

Estimating the Minimum Vertex Cover with an approximation factor of 7/5 for large enough undirected graphs encoded as a Boolean adjacency matrix 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 2
  -b, --bruteForce      enable comparison with the exponential-time brute-force approach
  -c, --count           calculate the size of the vertex cover
  -v, --verbose         enable 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_cover is provided for evaluating the Algorithm using randomly generated, large sparse matrices. It supports the following options:

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

The Capablanca Testing Application.

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 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         enable 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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Complexity

+ We present a polynomial-time algorithm achieving an approximation ratio of 7/5 for MVC, providing strong evidence that P = NP by efficiently solving a computationally hard problem with near-optimal solutions.

+ This result contradicts the Unique Games Conjecture, suggesting that many optimization problems may admit better solutions, revolutionizing theoretical computer science.

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License

• MIT License.