$1-$ The $2-$center problem is a complex problem that cannot be solved in polynomial time by an enumerative algorithm due to its combinatorial nature.

$2-$ The Branch$-$and$-$Bound Cut Algorithm is a method used for solving optimization problems, and it can be adapted to run in polynomial time. However, the actual running time depends on the specific problem and the quality of the bounds.

$3-$ The SIMPLEX algorithm is not a polynomial time algorithm. It is an exponential time algorithm. However, in practice, it often behaves like a polynomial time algorithm for many problems.

$4-$ The product of two binary variables is a non$-$linear operation. However, it can be linearized using additional variables and constraints.

$5-$ When the size of the bag is fixed, the knapsack problem can indeed be solved in polynomial time using dynamic programming. This is because the problem can be broken down into smaller subproblems, each of which can be solved in polynomial time.

$6-$ The greedy algorithms discussed in the scope of IE $343$ are indeed improvement algorithms. They make the best choice at each decision point with the hope that these local optimums will lead to a global optimum.

$7-$ Stochastic programming is a field of optimization that deals with problems in which some of the parameters are uncertain. It is based on the concept of optimizing under uncertainty.

$8-$ The mathematical model for the Traveling Salesman Problem $($TSP$)$ may yield the optimal solution even when there are no subtour elimination constraints. However, the absence of these constraints can also lead to non$-$optimal solutions.

$9-$ Exact algorithms are algorithms that always find the optimal solution. However, they are not always feasible to implement in polynomial time due to their complexity. They are often used for problems where an approximate solution is not sufficient.