sagemathgh-37181: Fix subtour elimination constraints in longest_cycle

    
An issue has been raised (see
sagemath#37028 (comment)) on
the formulation used to find the longest (induced) cycle. This was due
to the subtour elimination constraints that were not correct. We change
these constraints to fix this issue. The new constraints force to use
edges from the boundary of a subtour only when a vertex of that subtour
is selected.

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->
<!-- If your change requires a documentation PR, please link it
appropriately -->
<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
<!-- Feel free to remove irrelevant items. -->

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies

<!-- List all open PRs that this PR logically depends on
- sagemath#12345: short description why this is a dependency
- sagemath#34567: ...
-->

<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
    
URL: sagemath#37181
Reported by: David Coudert
Reviewer(s): Travis Scrimshaw

src/sage/graphs/generic_graph.py

@@ -7976,6 +7976,17 @@ def longest_cycle(self, induced=False, use_edge_labels=False,
 
         TESTS:
 
+        Check that the example from :issue:`37028` is fixed::
+
+            sage: d = {0: [4, 6, 10, 11], 1: [5, 7, 10, 11], 2: [8, 9, 10, 11],
+            ....:      3: [8, 9, 11], 4: [6, 10, 11], 5: [7, 10, 11],
+            ....:      6: [10, 11],  7: [10], 8: [10], 9: [11]}
+            sage: Z = Graph(d)
+            sage: Z.longest_cycle()
+            longest cycle: Graph on 9 vertices
+            sage: Z.longest_cycle(induced=True)
+            longest induced cycle: Graph on 5 vertices
+
         Small cases::
 
             sage: Graph().longest_cycle()
@@ -8085,8 +8096,8 @@ def F(e):
                                       constraint_generation=True)
 
         # We need one binary variable per vertex and per edge
-        vertex = p.new_variable(binary=True)
-        edge = p.new_variable(binary=True)
+        vertex = p.new_variable(binary=True, name='vertex')
+        edge = p.new_variable(binary=True, name='edge')
 
         # Objective function: maximize the size of the cycle
         p.set_objective(p.sum(weight(e) * edge[F(e)] for e in G.edge_iterator()))
@@ -8165,12 +8176,14 @@ def F(e):
 
                 # Add subtour elimination constraints
                 if directed:
-                    p.add_constraint(p.sum(edge[F(e)] for e in G.edge_boundary(c)), min=1)
                     c = set(c)
                     cbar = (v for v in G if v not in c)
-                    p.add_constraint(p.sum(edge[F(e)] for e in G.edge_boundary(cbar, c)), min=1)
+                    for u in c:
+                        p.add_constraint(vertex[u] <= p.sum(edge[F(e)] for e in G.edge_boundary(c)))
+                        p.add_constraint(vertex[u] <= p.sum(edge[F(e)] for e in G.edge_boundary(cbar, c)))
                 else:
-                    p.add_constraint(p.sum(edge[F(e)] for e in G.edge_boundary(c)), min=2)
+                    for u in c:
+                        p.add_constraint(2*vertex[u] <= p.sum(edge[F(e)] for e in G.edge_boundary(c)))
 
                 if induced:
                     # We eliminate this cycle