|
| 1 | +""" |
| 2 | +* Author: Manuel Di Lullo (https://github.com/manueldilullo) |
| 3 | +* Description: Approximization algorithm for minimum vertex cover problem. |
| 4 | + Matching Approach. Uses graphs represented with an adjacency list |
| 5 | +
|
| 6 | +URL: https://mathworld.wolfram.com/MinimumVertexCover.html |
| 7 | +URL: https://www.princeton.edu/~aaa/Public/Teaching/ORF523/ORF523_Lec6.pdf |
| 8 | +""" |
| 9 | + |
| 10 | + |
| 11 | +def matching_min_vertex_cover(graph: dict) -> set: |
| 12 | + """ |
| 13 | + APX Algorithm for min Vertex Cover using Matching Approach |
| 14 | + @input: graph (graph stored in an adjacency list where each vertex |
| 15 | + is represented as an integer) |
| 16 | + @example: |
| 17 | + >>> graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]} |
| 18 | + >>> matching_min_vertex_cover(graph) |
| 19 | + {0, 1, 2, 4} |
| 20 | + """ |
| 21 | + # chosen_vertices = set of chosen vertices |
| 22 | + chosen_vertices = set() |
| 23 | + # edges = list of graph's edges |
| 24 | + edges = get_edges(graph) |
| 25 | + |
| 26 | + # While there are still elements in edges list, take an arbitrary edge |
| 27 | + # (from_node, to_node) and add his extremity to chosen_vertices and then |
| 28 | + # remove all arcs adjacent to the from_node and to_node |
| 29 | + while edges: |
| 30 | + from_node, to_node = edges.pop() |
| 31 | + chosen_vertices.add(from_node) |
| 32 | + chosen_vertices.add(to_node) |
| 33 | + for edge in edges.copy(): |
| 34 | + if from_node in edge or to_node in edge: |
| 35 | + edges.discard(edge) |
| 36 | + return chosen_vertices |
| 37 | + |
| 38 | + |
| 39 | +def get_edges(graph: dict) -> set: |
| 40 | + """ |
| 41 | + Return a set of couples that represents all of the edges. |
| 42 | + @input: graph (graph stored in an adjacency list where each vertex is |
| 43 | + represented as an integer) |
| 44 | + @example: |
| 45 | + >>> graph = {0: [1, 3], 1: [0, 3], 2: [0, 3], 3: [0, 1, 2]} |
| 46 | + >>> get_edges(graph) |
| 47 | + {(0, 1), (3, 1), (0, 3), (2, 0), (3, 0), (2, 3), (1, 0), (3, 2), (1, 3)} |
| 48 | + """ |
| 49 | + edges = set() |
| 50 | + for from_node, to_nodes in graph.items(): |
| 51 | + for to_node in to_nodes: |
| 52 | + edges.add((from_node, to_node)) |
| 53 | + return edges |
| 54 | + |
| 55 | + |
| 56 | +if __name__ == "__main__": |
| 57 | + import doctest |
| 58 | + |
| 59 | + doctest.testmod() |
| 60 | + |
| 61 | + # graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]} |
| 62 | + # print(f"Matching vertex cover:\n{matching_min_vertex_cover(graph)}") |
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