#### Matching Min Vertex Cover

M
```"""
* Author: Manuel Di Lullo (https://github.com/manueldilullo)
* Description: Approximization algorithm for minimum vertex cover problem.
Matching Approach. Uses graphs represented with an adjacency list

URL: https://mathworld.wolfram.com/MinimumVertexCover.html
URL: https://www.princeton.edu/~aaa/Public/Teaching/ORF523/ORF523_Lec6.pdf
"""

def matching_min_vertex_cover(graph: dict) -> set:
"""
APX Algorithm for min Vertex Cover using Matching Approach
@input: graph (graph stored in an adjacency list where each vertex
is represented as an integer)
@example:
>>> graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
>>> matching_min_vertex_cover(graph)
{0, 1, 2, 4}
"""
# chosen_vertices = set of chosen vertices
chosen_vertices = set()
# edges = list of graph's edges
edges = get_edges(graph)

# While there are still elements in edges list, take an arbitrary edge
# (from_node, to_node) and add his extremity to chosen_vertices and then
# remove all arcs adjacent to the from_node and to_node
while edges:
from_node, to_node = edges.pop()
for edge in edges.copy():
if from_node in edge or to_node in edge:
return chosen_vertices

def get_edges(graph: dict) -> set:
"""
Return a set of couples that represents all of the edges.
@input: graph (graph stored in an adjacency list where each vertex is
represented as an integer)
@example:
>>> graph = {0: [1, 3], 1: [0, 3], 2: [0, 3], 3: [0, 1, 2]}
>>> get_edges(graph)
{(0, 1), (3, 1), (0, 3), (2, 0), (3, 0), (2, 3), (1, 0), (3, 2), (1, 3)}
"""
edges = set()
for from_node, to_nodes in graph.items():
for to_node in to_nodes:
return edges

if __name__ == "__main__":
import doctest

doctest.testmod()

# graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
# print(f"Matching vertex cover:\n{matching_min_vertex_cover(graph)}")
```  