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+
+Overview
+The Graph Optimization Project provides implementations of various graph algorithms, including Minimum Spanning Tree (MST) and Traveling Salesman Problem (TSP). This project showcases the use of both C++ and Python to solve these problems, leveraging different algorithmic approaches and libraries.
+
+##
+Features
+- **Minimum Spanning Tree (MST) Algorithms**:
+  - Boruvka's Algorithm
+  - Prim's Algorithm
+- **Traveling Salesman Problem (TSP) Solution**:
+  - Double TSP approximation technique
+  - Minimum-Cost Perfect Matching using Integer Programming
+- **Graph Representations and Utilities**:
+  - Union-Find data structure for cycle detection and union operations
+
+##
+Installation Instructions
+
+### Prerequisites
+- **C++ Compiler** (e.g., `g++`)
+- **Python 3.x**
+- **Gurobi Optimizer** (Python library)
+- **Git** to clone the repository
+
+### Steps
+1. **Clone the Repository**
+    ```bash
+    git clone https://github.com/leotac/graph.git
+    cd graph
+    ```
+
+2. **C++ Compilation**
+    - Ensure you have a C++ compiler installed.
+    - Compile the C++ files:
+    ```bash
+    g++ -o graph main.cpp
+    ```
+
+3. **Python Setup**
+    - Ensure you have Python 3 installed.
+    - Install the Gurobi library (Make sure you have a valid Gurobi license):
+    ```bash
+    pip install gurobipy
+    ```
+
+##
+Usage Examples
+
+### Running the C++ Implementation
+```bash
+./graph
+```
+
+### Using the Python Implementation
+```python
+import mst
+import tsp
+
+nodes = [0, 1, 2, 3, 4, 5]
+cost = [
+    [0, 0, 3.2, 2.2, 0, 0],
+    [0, 0, 2, 4, 0, 0],
+    [3.2, 2, 0, 3.1, 0, 0],
+    [2.2, 4, 3.1, 0, 3, 0],
+    [0, 0, 0, 3, 0, 1],
+    [0, 0, 0, 0, 1, 0]
+]
+
+# Solve MST using Python
+mst_tree = mst.kruskal(nodes, cost)
+print("MST:", mst_tree)
+
+# Solve TSP using Python
+tsp_tour = tsp.double_tsp(nodes, cost)
+print("TSP Tour:", tsp_tour)
+```
+
+##
+Code Summary
+
+- **main.cpp**: Entry point for the C++ implementation. Defines the cost matrix and initializes the algorithms.
+- **mst.cpp**: Contains the C++ implementation for MST algorithms using Boruvka's and Prim's methods.
+- **mst.py**: Contains the Python implementation for MST using Kruskal's algorithm.
+- **tsp.cpp**: Implements a double TSP approximation technique in C++.
+- **tsp.py**: Implements TSP approximation and minimum-cost perfect matching using Integer Programming via Gurobi.
+
+##
+Contributing Guidelines
+We welcome contributions to improve and expand the project. To contribute:
+
+1. Fork the repository.
+2. Create a new branch for your feature or bugfix.
+3. Commit your changes and push the branch to your fork.
+4. Create a pull request with a description of your changes.
+
+Please ensure that your code adheres to the existing code style and includes tests where appropriate.
+
+##
+License
+This project is licensed under the MIT License. See the [LICENSE](LICENSE) file for more details.
+```
+
+### Important Note
+Replace the placeholder `LICENSE` with the actual path to your license file in the repository. If there is no license file, add one using the text of the MIT License or whichever you prefer.
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