Greedy min vertex cover hacktoberfest (#5241)

* added complete graph generator function

* added doctest, type hints, wikipedia explanation

* added return type hint for function complete_graph

* added descriptive name for the parameter: n

* random graph generator with doctest and type hints

* added Greedy min vertex algorithm

* pre-commit hook(s) made changes

* Delete complete_graph_generator.py

* Delete random_graph_generator.py

* fixed doctest

* updated commit following highligths

* fixed following pre-commit highlights

* modified variables names

graphs/greedy_min_vertex_cover.py

@@ -0,0 +1,65 @@
+"""
+* Author: Manuel Di Lullo (https://github.com/manueldilullo)
+* Description: Approximization algorithm for minimum vertex cover problem.
+               Greedy Approach. Uses graphs represented with an adjacency list
+
+URL: https://mathworld.wolfram.com/MinimumVertexCover.html
+URL: https://cs.stackexchange.com/questions/129017/greedy-algorithm-for-vertex-cover
+"""
+
+import heapq
+
+
+def greedy_min_vertex_cover(graph: dict) -> set:
+    """
+    Greedy APX Algorithm for min Vertex Cover
+    @input: graph (graph stored in an adjacency list where each vertex
+            is represented with an integer)
+    @example:
+    >>> graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
+    >>> greedy_min_vertex_cover(graph)
+    {0, 1, 2, 4}
+    """
+    # queue used to store nodes and their rank
+    queue = []
+
+    # for each node and his adjacency list add them and the rank of the node to queue
+    # using heapq module the queue will be filled like a Priority Queue
+    # heapq works with a min priority queue, so I used -1*len(v) to build it
+    for key, value in graph.items():
+        # O(log(n))
+        heapq.heappush(queue, [-1 * len(value), (key, value)])
+
+    # chosen_vertices = set of chosen vertices
+    chosen_vertices = set()
+
+    # while queue isn't empty and there are still edges
+    #   (queue[0][0] is the rank of the node with max rank)
+    while queue and queue[0][0] != 0:
+        # extract vertex with max rank from queue and add it to chosen_vertices
+        argmax = heapq.heappop(queue)[1][0]
+        chosen_vertices.add(argmax)
+
+        # Remove all arcs adjacent to argmax
+        for elem in queue:
+            # if v haven't adjacent node, skip
+            if elem[0] == 0:
+                continue
+            # if argmax is reachable from elem
+            # remove argmax from elem's adjacent list and update his rank
+            if argmax in elem[1][1]:
+                index = elem[1][1].index(argmax)
+                del elem[1][1][index]
+                elem[0] += 1
+        # re-order the queue
+        heapq.heapify(queue)
+    return chosen_vertices
+
+
+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"Minimum vertex cover:\n{greedy_min_vertex_cover(graph)}")