baldor 0.1.4
Solve the Approximate Minimum Dominating Set for undirected graph encoded in DIMACS format.
License: MIT License
Requires Python: >=3.10

# Baldor: Approximate Minimum Dominating Set Solver

![Honoring the Memory of Aurelio Angel Baldor de la Vega (Cuban mathematician, educator and lawyer)](docs/baldor.jpg)

This work builds upon [Bipartite-Based 2-Approximation for Dominating Sets in General Graphs](https://dev.to/frank_vega_987689489099bf/bipartite-based-2-approximation-for-dominating-sets-in-general-graphs-l5g).

---

# Overview of the Minimum Dominating Set (MDS)

## Definition:

A **dominating set** in a graph $G = (V, E)$ is a subset $D \subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The **minimum dominating set (MDS)** is the smallest possible dominating set in terms of the number of vertices.

## Key Concepts:

1. **Graph Representation**:

   - $V$: Set of vertices.
   - $E$: Set of edges connecting the vertices.

2. **Dominating Set**:

   - A set $D$ where for every vertex $v \in V$, either $v \in D$ or $v$ is adjacent to some vertex in $D$.

3. **Minimum Dominating Set**:
   - The dominating set with the smallest cardinality (i.e., the fewest number of vertices).

## Applications:

- **Network Design**: Ensuring coverage in wireless sensor networks.
- **Social Networks**: Identifying influential nodes.
- **Game Theory**: Strategies in certain types of games.
- **Biology**: Modeling protein-protein interaction networks.

## Computational Complexity:

- **NP-Hard**: Finding the minimum dominating set is computationally intensive for large graphs.
- **Approximation Algorithms**: Used to find near-optimal solutions in polynomial time.

## Algorithms:

1. **Greedy Algorithm**:

   - Iteratively selects the vertex that covers the most uncovered vertices.
   - Provides a logarithmic approximation ratio.

2. **Integer Linear Programming (ILP)**:

   - Formulates the problem as an optimization problem.
   - Solvable using ILP solvers for exact solutions, though computationally expensive.

3. **Heuristics and Metaheuristics**:
   - Genetic algorithms, simulated annealing, etc., for large-scale problems.

## Challenges:

- **Scalability**: Exact algorithms are infeasible for very large graphs.
- **Dynamic Graphs**: Maintaining a minimum dominating set in graphs that change over time.

## Research Directions:

- **Parallel Algorithms**: Leveraging multi-core processors and distributed computing.
- **Machine Learning**: Using learning-based approaches to predict dominating sets.
- **Hybrid Methods**: Combining exact and heuristic methods for better performance.

## Conclusion:

The minimum dominating set problem is a fundamental issue in graph theory with wide-ranging applications. While it is computationally challenging, various algorithms and heuristics provide practical solutions for different scenarios. Ongoing research continues to improve the efficiency and applicability of these methods.

---

# Algorithm Overview: Baldor for Minimum Dominating Set

## Overview

Baldor is an approximate algorithm for the minimum dominating set problem on general graphs. It constructs a bipartite graph $B$ from the input graph $G = (V, E)$, where $V_B = \{ (v, 0), (v, 1) \mid v \in V \}$, and uses a greedy approach to select vertices that dominate all pairs $(u, 0)$ and $(u, 1)$. The algorithm guarantees a 2-approximation by ensuring that each vertex $u \in V$ contributes at most two vertices to the dominating set $S$.

## Runtime

- **Complexity:** The algorithm's runtime is $O(|V| \cdot |E|)$, where $|V|$ is the number of vertices and $|E|$ is the number of edges in $G$. This arises from constructing the bipartite graph $B$ and performing a greedy domination step.
- **Experimental Performance:** On _Second DIMACS Implementation Challenge_ instances:
  - Small graphs (e.g., `san200_0.7_1.clq`): 93.572 ms
  - Larger graphs (e.g., `san1000.clq`): 1959.679 ms
  - The runtime scales with graph size but remains competitive with NetworkX, which is faster on small graphs.

## Correctness

- **Approximation Guarantee:** Baldor achieves a 2-approximation for the minimum dominating set. This is proven by bounding $|D_u| \leq 2$ for each vertex $u$, where $D_u$ is the set of vertices in $S$ dominating $(u, 0)$ and $(u, 1)$ in $B$. The bipartite construction and greedy selection ensure all vertices are dominated.
- **Experimental Validation:** The approximation quality metric often approaches 2, confirming near-optimal performance. On `san1000.clq`, Baldor yields a set of size 4 versus NetworkX’s 40, indicating significant improvement.

---

## Problem Statement

Input: A Boolean Adjacency Matrix $M$.

Answer: Find a Minimum Dominating Set.

### 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](http://dimacs.rutgers.edu/Challenges) 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:_

Dominating Set Found `4, 5`: Nodes `4` and `5` constitute an optimal solution.

---

# Compile and Environment

## Prerequisites

- Python ≥ 3.10

## Installation

```bash
pip install baldor
```

## Execution

1. Clone the repository:

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

2. Run the script:

   ```bash
   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` compressed text files.

   **Example Output:**

   ```
   testMatrix1: Dominating Set Found 4, 5
   ```

   This indicates nodes `4, 5` form a Dominating Set.

---

## Dominating Set Size

Use the `-c` flag to count the nodes in the Dominating Set:

```bash
solve -i ./benchmarks/testMatrix2 -c
```

**Output:**

```
testMatrix2: Dominating Set Size 2
```

---

# Command Options

Display help and options:

```bash
solve -h
```

**Output:**

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

Solve the Approximate Minimum Dominating Set 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 logarithmic factor
  -b, --bruteForce      enable comparison with the exponential-time brute-force approach
  -c, --count           calculate the size of the Dominating Set
  -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_solve` command, use the following in your terminal or command prompt:

```bash
batch_solve -h
```

This will display the following help information:

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

Solve the Approximate Minimum Dominating Set 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 logarithmic factor
  -b, --bruteForce      enable comparison with the exponential-time brute-force approach
  -c, --count           calculate the size of the Dominating Set
  -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_solve` is provided for evaluating the Algorithm using randomly generated, large sparse matrices. It supports the following options:

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

The Baldor 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 logarithmic factor
  -b, --bruteForce      enable comparison with the exponential-time brute-force approach
  -c, --count           calculate the size of the Dominating Set
  -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**.

---

# Complexity

```diff
+ We present a polynomial-time algorithm achieving a 2-approximation ratio for MDS, providing strong evidence that P = NP by efficiently solving a computationally hard problem with near-optimal solutions.
```

---

# License

- MIT License.