Baldor: Approximate Minimum Dominating Set Solver

This work builds upon Bipartite-Based 2-Approximation for Dominating Sets in General Graphs.

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

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:

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

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 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:

solve -i ./benchmarks/testMatrix2 -c

Output:

testMatrix2: Dominating Set Size 2

Command Options

Display help and options:

solve -h

Output:

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:

batch_solve -h

This will display the following help information:

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:

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

+ 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.