Matching min vertex cover (#5326)

* matching algorithm for min vertex cover problem

* fixed hint on row 37

* changed variable names

* provided doctest for get_edges function

* Removed dict.keys() iteration

* Update matching_min_vertex_cover.py
Co-authored-by: NChristian Clauss <cclauss@me.com>

graphs/matching_min_vertex_cover.py

+"""
+* 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()
+        chosen_vertices.add(from_node)
+        chosen_vertices.add(to_node)
+        for edge in edges.copy():
+            if from_node in edge or to_node in edge:
+                edges.discard(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:
+            edges.add((from_node, to_node))
+    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)}")