feat(backtracking): add m-coloring algorithm compatible with CI tests

backtracking/m_coloring_problem.py

@@ -0,0 +1,60 @@
+def is_safe(node: int, color: int, graph: list[list[int]], num_vertices: int,
+            col: list[int]) -> bool:
+    """
+    Check if it is safe to assign a color to a node.
+
+    >>> is_safe(0, 1, [[0,1],[1,0]], 2, [0,1])
+    False
+    >>> is_safe(0, 2, [[0,1],[1,0]], 2, [0,1])
+    True
+    """
+    return all(not (graph[node][k] == 1 and col[k] == color)
+               for k in range(num_vertices))
+
+
+def solve(node: int, col: list[int], max_colors: int, num_vertices: int,
+          graph: list[list[int]]) -> bool:
+    """
+    Recursively try to color the graph using at most max_colors.
+
+    >>> solve(0, [0]*3, 3, 3, [[0,1,0],[1,0,1],[0,1,0]])
+    True
+    >>> solve(0, [0]*3, 2, 3, [[0,1,0],[1,0,1],[0,1,0]])
+    False
+    """
+    if node == num_vertices:
+        return True
+    for c in range(1, max_colors + 1):
+        if is_safe(node, c, graph, num_vertices, col):
+            col[node] = c
+            if solve(node + 1, col, max_colors, num_vertices, graph):
+                return True
+            col[node] = 0
+    return False
+
+
+def graph_coloring(graph: list[list[int]], max_colors: int,
+                   num_vertices: int) -> bool:
+    """
+    Determine if the graph can be colored with at most max_colors.
+
+    >>> graph_coloring([[0,1,1],[1,0,1],[1,1,0]], 3, 3)
+    True
+    >>> graph_coloring([[0,1,1],[1,0,1],[1,1,0]], 2, 3)
+    False
+    """
+    col = [0] * num_vertices
+    return solve(0, col, max_colors, num_vertices, graph)
+
+
+if __name__ == "__main__":
+    num_vertices = int(input())
+    num_edges = int(input())
+    graph = [[0]*num_vertices for _ in range(num_vertices)]
+    for _ in range(num_edges):
+        u, v = map(int, input().split())
+        graph[u][v] = 1
+        graph[v][u] = 1
+    max_colors = int(input())
+    col = [0]*num_vertices
+    print(solve(0, col, max_colors, num_vertices, graph))