gh-40510: Improve Feng, PNC algorithm
    
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+ Fix several bugs of PNC algorithm.
+ Improve implementation of Feng algorithm.
+ Add PNC as a supported algorithm in shortest_simple_paths

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### ⌛ Dependencies

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#40364
    
URL: #40510
Reported by: Yuta Inoue
Reviewer(s): David Coudert

Author: Release Manager

src/sage/graphs/cycle_enumeration.py

@@ -265,11 +265,11 @@ def _all_simple_cycles_iterator_edge(self, edge, max_length=None,
         sage: g = graphs.Grid2dGraph(2, 5).to_directed()
         sage: it = g._all_simple_cycles_iterator_edge(((0, 0), (0, 1), None), report_weight=True)
         sage: for i in range(5): print(next(it))
-        (2, [(0, 0), (0, 1), (0, 0)])
-        (4, [(0, 0), (0, 1), (1, 1), (1, 0), (0, 0)])
-        (6, [(0, 0), (0, 1), (0, 2), (1, 2), (1, 1), (1, 0), (0, 0)])
-        (8, [(0, 0), (0, 1), (0, 2), (0, 3), (1, 3), (1, 2), (1, 1), (1, 0), (0, 0)])
-        (10, [(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (1, 4), (1, 3), (1, 2), (1, 1), (1, 0), (0, 0)])
+        (2.0, [(0, 0), (0, 1), (0, 0)])
+        (4.0, [(0, 0), (0, 1), (1, 1), (1, 0), (0, 0)])
+        (6.0, [(0, 0), (0, 1), (0, 2), (1, 2), (1, 1), (1, 0), (0, 0)])
+        (8.0, [(0, 0), (0, 1), (0, 2), (0, 3), (1, 3), (1, 2), (1, 1), (1, 0), (0, 0)])
+        (10.0, [(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (1, 4), (1, 3), (1, 2), (1, 1), (1, 0), (0, 0)])
 
     The function works for undirected graphs as well::
 
@@ -841,12 +841,12 @@ def all_simple_cycles(self, starting_vertices=None, rooted=False,
         sage: cycles_B = g.all_simple_cycles(weight_function=lambda e:e[0]+e[1], by_weight=True,
         ....:                                report_weight=True, algorithm='B')
         sage: cycles_B
-        [(2, [0, 1, 0]), (4, [0, 2, 0]), (6, [0, 1, 2, 0]), (6, [0, 2, 1, 0]),
-         (6, [0, 3, 0]), (6, [1, 2, 1]), (8, [0, 1, 3, 0]), (8, [0, 3, 1, 0]),
-         (8, [1, 3, 1]), (10, [0, 2, 3, 0]), (10, [0, 3, 2, 0]), (10, [2, 3, 2]),
-         (12, [0, 1, 3, 2, 0]), (12, [0, 1, 2, 3, 0]), (12, [0, 2, 3, 1, 0]),
-         (12, [0, 2, 1, 3, 0]), (12, [0, 3, 2, 1, 0]), (12, [0, 3, 1, 2, 0]),
-         (12, [1, 2, 3, 1]), (12, [1, 3, 2, 1])]
+        [(2.0, [0, 1, 0]), (4.0, [0, 2, 0]), (6.0, [0, 1, 2, 0]), (6.0, [0, 2, 1, 0]),
+         (6.0, [0, 3, 0]), (6.0, [1, 2, 1]), (8.0, [0, 1, 3, 0]), (8.0, [0, 3, 1, 0]),
+         (8.0, [1, 3, 1]), (10.0, [0, 2, 3, 0]), (10.0, [0, 3, 2, 0]), (10.0, [2, 3, 2]),
+         (12.0, [0, 1, 3, 2, 0]), (12.0, [0, 1, 2, 3, 0]), (12.0, [0, 2, 3, 1, 0]),
+         (12.0, [0, 2, 1, 3, 0]), (12.0, [0, 3, 2, 1, 0]), (12.0, [0, 3, 1, 2, 0]),
+         (12.0, [1, 2, 3, 1]), (12.0, [1, 3, 2, 1])]
         sage: cycles.sort() == cycles_B.sort()
         True